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Properties

Label 10-2842e5-1.1-c1e5-0-7
Degree $10$
Conductor $1.854\times 10^{17}$
Sign $-1$
Analytic cond. $6.01874\times 10^{6}$
Root an. cond. $4.76376$
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  + 5·2-s − 3·3-s + 15·4-s − 7·5-s − 15·6-s + 35·8-s + 9-s − 35·10-s − 45·12-s − 10·13-s + 21·15-s + 70·16-s − 8·17-s + 5·18-s − 2·19-s − 105·20-s + 23-s − 105·24-s + 18·25-s − 50·26-s + 4·27-s − 5·29-s + 105·30-s − 11·31-s + 126·32-s − 40·34-s + 15·36-s + ⋯
L(s)  = 1  + 3.53·2-s − 1.73·3-s + 15/2·4-s − 3.13·5-s − 6.12·6-s + 12.3·8-s + 1/3·9-s − 11.0·10-s − 12.9·12-s − 2.77·13-s + 5.42·15-s + 35/2·16-s − 1.94·17-s + 1.17·18-s − 0.458·19-s − 23.4·20-s + 0.208·23-s − 21.4·24-s + 18/5·25-s − 9.80·26-s + 0.769·27-s − 0.928·29-s + 19.1·30-s − 1.97·31-s + 22.2·32-s − 6.85·34-s + 5/2·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 7^{10} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{5} \cdot 7^{10} \cdot 29^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(2^{5} \cdot 7^{10} \cdot 29^{5}\)
Sign: $-1$
Analytic conductor: \(6.01874\times 10^{6}\)
Root analytic conductor: \(4.76376\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 2^{5} \cdot 7^{10} \cdot 29^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad2$C_1$ \( ( 1 - T )^{5} \)
7 \( 1 \)
29$C_1$ \( ( 1 + T )^{5} \)
good3$C_2 \wr S_5$ \( 1 + p T + 8 T^{2} + 17 T^{3} + 11 p T^{4} + 55 T^{5} + 11 p^{2} T^{6} + 17 p^{2} T^{7} + 8 p^{3} T^{8} + p^{5} T^{9} + p^{5} T^{10} \)
5$C_2 \wr S_5$ \( 1 + 7 T + 31 T^{2} + 21 p T^{3} + 293 T^{4} + 691 T^{5} + 293 p T^{6} + 21 p^{3} T^{7} + 31 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 2 p T^{2} - 54 T^{3} + 109 T^{4} - 1179 T^{5} + 109 p T^{6} - 54 p^{2} T^{7} + 2 p^{4} T^{8} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 10 T + 93 T^{2} + 513 T^{3} + 2664 T^{4} + 9855 T^{5} + 2664 p T^{6} + 513 p^{2} T^{7} + 93 p^{3} T^{8} + 10 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 8 T + 94 T^{2} + 522 T^{3} + 3329 T^{4} + 13121 T^{5} + 3329 p T^{6} + 522 p^{2} T^{7} + 94 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 2 T + 32 T^{2} + 14 T^{3} + 757 T^{4} + 711 T^{5} + 757 p T^{6} + 14 p^{2} T^{7} + 32 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 - T + 76 T^{2} - 189 T^{3} + 2555 T^{4} - 7603 T^{5} + 2555 p T^{6} - 189 p^{2} T^{7} + 76 p^{3} T^{8} - p^{4} T^{9} + p^{5} T^{10} \)
31$C_2 \wr S_5$ \( 1 + 11 T + 131 T^{2} + 1073 T^{3} + 8071 T^{4} + 44997 T^{5} + 8071 p T^{6} + 1073 p^{2} T^{7} + 131 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 + 8 T + 138 T^{2} + 762 T^{3} + 8211 T^{4} + 34833 T^{5} + 8211 p T^{6} + 762 p^{2} T^{7} + 138 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 23 T + 337 T^{2} + 3477 T^{3} + 29189 T^{4} + 200675 T^{5} + 29189 p T^{6} + 3477 p^{2} T^{7} + 337 p^{3} T^{8} + 23 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 3 T + 52 T^{2} + 137 T^{3} + 3085 T^{4} + 17531 T^{5} + 3085 p T^{6} + 137 p^{2} T^{7} + 52 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 16 T + 244 T^{2} + 2424 T^{3} + 22157 T^{4} + 157105 T^{5} + 22157 p T^{6} + 2424 p^{2} T^{7} + 244 p^{3} T^{8} + 16 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 - 7 T + 208 T^{2} - 1533 T^{3} + 19187 T^{4} - 122989 T^{5} + 19187 p T^{6} - 1533 p^{2} T^{7} + 208 p^{3} T^{8} - 7 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 9 T + 256 T^{2} - 1833 T^{3} + 28669 T^{4} - 154293 T^{5} + 28669 p T^{6} - 1833 p^{2} T^{7} + 256 p^{3} T^{8} - 9 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 15 T + 295 T^{2} + 3022 T^{3} + 577 p T^{4} + 262381 T^{5} + 577 p^{2} T^{6} + 3022 p^{2} T^{7} + 295 p^{3} T^{8} + 15 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 + 4 T + 161 T^{2} + 367 T^{3} + 14278 T^{4} + 24747 T^{5} + 14278 p T^{6} + 367 p^{2} T^{7} + 161 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 22 T + 406 T^{2} + 5106 T^{3} + 58901 T^{4} + 521437 T^{5} + 58901 p T^{6} + 5106 p^{2} T^{7} + 406 p^{3} T^{8} + 22 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 118 T^{2} + 226 T^{3} + 7843 T^{4} + 33337 T^{5} + 7843 p T^{6} + 226 p^{2} T^{7} + 118 p^{3} T^{8} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 + 13 T + 362 T^{2} + 3727 T^{3} + 54655 T^{4} + 428163 T^{5} + 54655 p T^{6} + 3727 p^{2} T^{7} + 362 p^{3} T^{8} + 13 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 + 28 T + 628 T^{2} + 8973 T^{3} + 112748 T^{4} + 1077883 T^{5} + 112748 p T^{6} + 8973 p^{2} T^{7} + 628 p^{3} T^{8} + 28 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 17 T + 250 T^{2} + 2316 T^{3} + 32888 T^{4} + 305705 T^{5} + 32888 p T^{6} + 2316 p^{2} T^{7} + 250 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 42 T + 1024 T^{2} + 18106 T^{3} + 247831 T^{4} + 2707201 T^{5} + 247831 p T^{6} + 18106 p^{2} T^{7} + 1024 p^{3} T^{8} + 42 p^{4} T^{9} + p^{5} T^{10} \)
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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

−5.49309309228463545295925437027, −5.26472880568909878577360395396, −5.21487319081941694376704900604, −5.18074850931924726154974558280, −4.94637061191778016761641610990, −4.88098556482957925673773739166, −4.55402668345791104712431028735, −4.40402726620414147595580142323, −4.40341404326942714998195828219, −4.30117814300836855188890856755, −4.04321432478185671176907791488, −3.78533211784464000497506071189, −3.63859373649584479674640626054, −3.57771181879290227296319645853, −3.47659756898704189107202459999, −3.12468921968438183811165060433, −2.99206435617086103305650882671, −2.70915382162009503824310560576, −2.62288994853444808375162110032, −2.51790914657472791020462889923, −1.98700655484632523142578846600, −1.98374511470141636639797059169, −1.62230202159274976889338111277, −1.47489284024131066636424286719, −1.27612471296597777331642573794, 0, 0, 0, 0, 0, 1.27612471296597777331642573794, 1.47489284024131066636424286719, 1.62230202159274976889338111277, 1.98374511470141636639797059169, 1.98700655484632523142578846600, 2.51790914657472791020462889923, 2.62288994853444808375162110032, 2.70915382162009503824310560576, 2.99206435617086103305650882671, 3.12468921968438183811165060433, 3.47659756898704189107202459999, 3.57771181879290227296319645853, 3.63859373649584479674640626054, 3.78533211784464000497506071189, 4.04321432478185671176907791488, 4.30117814300836855188890856755, 4.40341404326942714998195828219, 4.40402726620414147595580142323, 4.55402668345791104712431028735, 4.88098556482957925673773739166, 4.94637061191778016761641610990, 5.18074850931924726154974558280, 5.21487319081941694376704900604, 5.26472880568909878577360395396, 5.49309309228463545295925437027

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