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Properties

Label 10-2299e5-1.1-c1e5-0-0
Degree $10$
Conductor $6.422\times 10^{16}$
Sign $-1$
Analytic cond. $2.08487\times 10^{6}$
Root an. cond. $4.28457$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $5$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s − 5·5-s − 2·6-s − 6·7-s + 2·8-s − 5·9-s + 10·10-s − 4·13-s + 12·14-s − 5·15-s − 16-s + 4·17-s + 10·18-s + 5·19-s − 6·21-s + 3·23-s + 2·24-s + 3·25-s + 8·26-s − 10·27-s − 10·29-s + 10·30-s + 11·31-s − 8·34-s + 30·35-s + 37-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s − 2.23·5-s − 0.816·6-s − 2.26·7-s + 0.707·8-s − 5/3·9-s + 3.16·10-s − 1.10·13-s + 3.20·14-s − 1.29·15-s − 1/4·16-s + 0.970·17-s + 2.35·18-s + 1.14·19-s − 1.30·21-s + 0.625·23-s + 0.408·24-s + 3/5·25-s + 1.56·26-s − 1.92·27-s − 1.85·29-s + 1.82·30-s + 1.97·31-s − 1.37·34-s + 5.07·35-s + 0.164·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{10} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{10} \cdot 19^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(11^{10} \cdot 19^{5}\)
Sign: $-1$
Analytic conductor: \(2.08487\times 10^{6}\)
Root analytic conductor: \(4.28457\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 11^{10} \cdot 19^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad11 \( 1 \)
19$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p T + p^{2} T^{2} + 3 p T^{3} + 9 T^{4} + 3 p^{2} T^{5} + 9 p T^{6} + 3 p^{3} T^{7} + p^{5} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 - T + 2 p T^{2} - T^{3} + 16 T^{4} + 11 T^{5} + 16 p T^{6} - p^{2} T^{7} + 2 p^{4} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + p T + 22 T^{2} + 67 T^{3} + 196 T^{4} + 439 T^{5} + 196 p T^{6} + 67 p^{2} T^{7} + 22 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 6 T + 34 T^{2} + 106 T^{3} + 50 p T^{4} + 832 T^{5} + 50 p^{2} T^{6} + 106 p^{2} T^{7} + 34 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 4 T + 56 T^{2} + 14 p T^{3} + 1376 T^{4} + 3378 T^{5} + 1376 p T^{6} + 14 p^{3} T^{7} + 56 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 - 4 T + 53 T^{2} - 208 T^{3} + 1562 T^{4} - 4696 T^{5} + 1562 p T^{6} - 208 p^{2} T^{7} + 53 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - 3 T + 39 T^{2} + 112 T^{3} - 178 T^{4} + 7542 T^{5} - 178 p T^{6} + 112 p^{2} T^{7} + 39 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 10 T + 108 T^{2} + 504 T^{3} + 4 p^{2} T^{4} + 11922 T^{5} + 4 p^{3} T^{6} + 504 p^{2} T^{7} + 108 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 - 11 T + 152 T^{2} - 1171 T^{3} + 300 p T^{4} - 52217 T^{5} + 300 p^{2} T^{6} - 1171 p^{2} T^{7} + 152 p^{3} T^{8} - 11 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - T + 105 T^{2} - 44 T^{3} + 6330 T^{4} - 3606 T^{5} + 6330 p T^{6} - 44 p^{2} T^{7} + 105 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 2 T + 16 T^{2} + 76 T^{3} + 816 T^{4} + 3620 T^{5} + 816 p T^{6} + 76 p^{2} T^{7} + 16 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 20 T + 238 T^{2} + 1800 T^{3} + 11614 T^{4} + 1618 p T^{5} + 11614 p T^{6} + 1800 p^{2} T^{7} + 238 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 20 T + 263 T^{2} + 2672 T^{3} + 23846 T^{4} + 175992 T^{5} + 23846 p T^{6} + 2672 p^{2} T^{7} + 263 p^{3} T^{8} + 20 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 14 T + 177 T^{2} + 1576 T^{3} + 15906 T^{4} + 118708 T^{5} + 15906 p T^{6} + 1576 p^{2} T^{7} + 177 p^{3} T^{8} + 14 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 3 T + 131 T^{2} + 200 T^{3} + 5286 T^{4} + 42486 T^{5} + 5286 p T^{6} + 200 p^{2} T^{7} + 131 p^{3} T^{8} - 3 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 - 10 T + 281 T^{2} - 1976 T^{3} + 31554 T^{4} - 165916 T^{5} + 31554 p T^{6} - 1976 p^{2} T^{7} + 281 p^{3} T^{8} - 10 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 9 T + 140 T^{2} - 1585 T^{3} + 16328 T^{4} - 113899 T^{5} + 16328 p T^{6} - 1585 p^{2} T^{7} + 140 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 - 23 T + 338 T^{2} - 3603 T^{3} + 32304 T^{4} - 260659 T^{5} + 32304 p T^{6} - 3603 p^{2} T^{7} + 338 p^{3} T^{8} - 23 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 25 T^{2} + 16 p T^{3} + 6558 T^{4} + 15136 T^{5} + 6558 p T^{6} + 16 p^{3} T^{7} + 25 p^{3} T^{8} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 44 T + 1143 T^{2} + 20032 T^{3} + 263862 T^{4} + 2652648 T^{5} + 263862 p T^{6} + 20032 p^{2} T^{7} + 1143 p^{3} T^{8} + 44 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 14 T + 346 T^{2} - 3406 T^{3} + 47606 T^{4} - 368596 T^{5} + 47606 p T^{6} - 3406 p^{2} T^{7} + 346 p^{3} T^{8} - 14 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 27 T + 713 T^{2} + 10780 T^{3} + 152718 T^{4} + 1491426 T^{5} + 152718 p T^{6} + 10780 p^{2} T^{7} + 713 p^{3} T^{8} + 27 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 - 15 T + 361 T^{2} - 3704 T^{3} + 58310 T^{4} - 473762 T^{5} + 58310 p T^{6} - 3704 p^{2} T^{7} + 361 p^{3} T^{8} - 15 p^{4} T^{9} + p^{5} T^{10} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{10} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.80987793145193828417157843917, −5.60198865550564607016521604867, −5.41443671192646621370831253209, −5.16090888062739913036023286989, −5.13912610347083354789489475271, −5.04815243825630333061861012987, −4.85711363781180229278229145803, −4.64856379336519166452380844919, −4.30411285662515979332953155986, −4.15735926549435282539471193927, −3.87581577508900554131457254532, −3.75551791987102176660949173556, −3.55714650059919658548916379749, −3.35340719165621444635374364515, −3.30512605211977651883884992439, −3.28020646486238580088965473081, −3.15382659376182640238487585044, −2.74499879619296728117552008766, −2.43962118419631432537512476567, −2.42747140967625438797621191451, −2.25423917546517488283456584322, −1.69603754640173567622886092797, −1.34265527918121425302683720678, −1.28444810351292988025262185240, −0.929178432015594279525258105925, 0, 0, 0, 0, 0, 0.929178432015594279525258105925, 1.28444810351292988025262185240, 1.34265527918121425302683720678, 1.69603754640173567622886092797, 2.25423917546517488283456584322, 2.42747140967625438797621191451, 2.43962118419631432537512476567, 2.74499879619296728117552008766, 3.15382659376182640238487585044, 3.28020646486238580088965473081, 3.30512605211977651883884992439, 3.35340719165621444635374364515, 3.55714650059919658548916379749, 3.75551791987102176660949173556, 3.87581577508900554131457254532, 4.15735926549435282539471193927, 4.30411285662515979332953155986, 4.64856379336519166452380844919, 4.85711363781180229278229145803, 5.04815243825630333061861012987, 5.13912610347083354789489475271, 5.16090888062739913036023286989, 5.41443671192646621370831253209, 5.60198865550564607016521604867, 5.80987793145193828417157843917

Graph of the $Z$-function along the critical line