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Properties

Label 10-775e5-1.1-c1e5-0-3
Degree $10$
Conductor $2.796\times 10^{14}$
Sign $-1$
Analytic cond. $9075.99$
Root an. cond. $2.48765$
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  − 4·2-s − 3-s + 6·4-s + 4·6-s − 6·7-s − 5·8-s − 6·9-s − 6·12-s − 4·13-s + 24·14-s + 8·16-s − 11·17-s + 24·18-s − 4·19-s + 6·21-s − 12·23-s + 5·24-s + 16·26-s + 8·27-s − 36·28-s − 6·29-s + 5·31-s − 17·32-s + 44·34-s − 36·36-s + 2·37-s + 16·38-s + ⋯
L(s)  = 1  − 2.82·2-s − 0.577·3-s + 3·4-s + 1.63·6-s − 2.26·7-s − 1.76·8-s − 2·9-s − 1.73·12-s − 1.10·13-s + 6.41·14-s + 2·16-s − 2.66·17-s + 5.65·18-s − 0.917·19-s + 1.30·21-s − 2.50·23-s + 1.02·24-s + 3.13·26-s + 1.53·27-s − 6.80·28-s − 1.11·29-s + 0.898·31-s − 3.00·32-s + 7.54·34-s − 6·36-s + 0.328·37-s + 2.59·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{10} \cdot 31^{5}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{5} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(10\)
Conductor: \(5^{10} \cdot 31^{5}\)
Sign: $-1$
Analytic conductor: \(9075.99\)
Root analytic conductor: \(2.48765\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(5\)
Selberg data: \((10,\ 5^{10} \cdot 31^{5} ,\ ( \ : 1/2, 1/2, 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad5 \( 1 \)
31$C_1$ \( ( 1 - T )^{5} \)
good2$C_2 \wr S_5$ \( 1 + p^{2} T + 5 p T^{2} + 21 T^{3} + 9 p^{2} T^{4} + 53 T^{5} + 9 p^{3} T^{6} + 21 p^{2} T^{7} + 5 p^{4} T^{8} + p^{6} T^{9} + p^{5} T^{10} \)
3$C_2 \wr S_5$ \( 1 + T + 7 T^{2} + 5 T^{3} + p^{3} T^{4} + 11 T^{5} + p^{4} T^{6} + 5 p^{2} T^{7} + 7 p^{3} T^{8} + p^{4} T^{9} + p^{5} T^{10} \)
7$C_2 \wr S_5$ \( 1 + 6 T + 36 T^{2} + 19 p T^{3} + 492 T^{4} + 1311 T^{5} + 492 p T^{6} + 19 p^{3} T^{7} + 36 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
11$C_2 \wr S_5$ \( 1 + 32 T^{2} - 37 T^{3} + 470 T^{4} - 777 T^{5} + 470 p T^{6} - 37 p^{2} T^{7} + 32 p^{3} T^{8} + p^{5} T^{10} \)
13$C_2 \wr S_5$ \( 1 + 4 T + 58 T^{2} + 185 T^{3} + 1438 T^{4} + 3467 T^{5} + 1438 p T^{6} + 185 p^{2} T^{7} + 58 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
17$C_2 \wr S_5$ \( 1 + 11 T + 98 T^{2} + 631 T^{3} + 199 p T^{4} + 15037 T^{5} + 199 p^{2} T^{6} + 631 p^{2} T^{7} + 98 p^{3} T^{8} + 11 p^{4} T^{9} + p^{5} T^{10} \)
19$C_2 \wr S_5$ \( 1 + 4 T + 72 T^{2} + 214 T^{3} + 121 p T^{4} + 5379 T^{5} + 121 p^{2} T^{6} + 214 p^{2} T^{7} + 72 p^{3} T^{8} + 4 p^{4} T^{9} + p^{5} T^{10} \)
23$C_2 \wr S_5$ \( 1 + 12 T + 146 T^{2} + 1024 T^{3} + 7163 T^{4} + 34575 T^{5} + 7163 p T^{6} + 1024 p^{2} T^{7} + 146 p^{3} T^{8} + 12 p^{4} T^{9} + p^{5} T^{10} \)
29$C_2 \wr S_5$ \( 1 + 6 T + 132 T^{2} + 626 T^{3} + 7309 T^{4} + 26351 T^{5} + 7309 p T^{6} + 626 p^{2} T^{7} + 132 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
37$C_2 \wr S_5$ \( 1 - 2 T + 15 T^{2} - 483 T^{3} + 2692 T^{4} - 2727 T^{5} + 2692 p T^{6} - 483 p^{2} T^{7} + 15 p^{3} T^{8} - 2 p^{4} T^{9} + p^{5} T^{10} \)
41$C_2 \wr S_5$ \( 1 + 2 T + 138 T^{2} + 177 T^{3} + 8458 T^{4} + 7763 T^{5} + 8458 p T^{6} + 177 p^{2} T^{7} + 138 p^{3} T^{8} + 2 p^{4} T^{9} + p^{5} T^{10} \)
43$C_2 \wr S_5$ \( 1 + 7 T + 75 T^{2} + 807 T^{3} + 6601 T^{4} + 31789 T^{5} + 6601 p T^{6} + 807 p^{2} T^{7} + 75 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
47$C_2 \wr S_5$ \( 1 + 8 T + 60 T^{2} + 222 T^{3} + 4777 T^{4} + 35803 T^{5} + 4777 p T^{6} + 222 p^{2} T^{7} + 60 p^{3} T^{8} + 8 p^{4} T^{9} + p^{5} T^{10} \)
53$C_2 \wr S_5$ \( 1 + 25 T + 463 T^{2} + 5821 T^{3} + 59401 T^{4} + 474857 T^{5} + 59401 p T^{6} + 5821 p^{2} T^{7} + 463 p^{3} T^{8} + 25 p^{4} T^{9} + p^{5} T^{10} \)
59$C_2 \wr S_5$ \( 1 - 4 T + 121 T^{2} - 459 T^{3} + 10062 T^{4} - 21719 T^{5} + 10062 p T^{6} - 459 p^{2} T^{7} + 121 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
61$C_2 \wr S_5$ \( 1 + 17 T + 282 T^{2} + 2283 T^{3} + 21613 T^{4} + 131403 T^{5} + 21613 p T^{6} + 2283 p^{2} T^{7} + 282 p^{3} T^{8} + 17 p^{4} T^{9} + p^{5} T^{10} \)
67$C_2 \wr S_5$ \( 1 - 13 T + 305 T^{2} - 2837 T^{3} + 37943 T^{4} - 263107 T^{5} + 37943 p T^{6} - 2837 p^{2} T^{7} + 305 p^{3} T^{8} - 13 p^{4} T^{9} + p^{5} T^{10} \)
71$C_2 \wr S_5$ \( 1 + 6 T + 288 T^{2} + 22 p T^{3} + 36889 T^{4} + 160763 T^{5} + 36889 p T^{6} + 22 p^{3} T^{7} + 288 p^{3} T^{8} + 6 p^{4} T^{9} + p^{5} T^{10} \)
73$C_2 \wr S_5$ \( 1 + 7 T + 306 T^{2} + 1644 T^{3} + 41248 T^{4} + 170925 T^{5} + 41248 p T^{6} + 1644 p^{2} T^{7} + 306 p^{3} T^{8} + 7 p^{4} T^{9} + p^{5} T^{10} \)
79$C_2 \wr S_5$ \( 1 - 12 T + 209 T^{2} - 1577 T^{3} + 17098 T^{4} - 114367 T^{5} + 17098 p T^{6} - 1577 p^{2} T^{7} + 209 p^{3} T^{8} - 12 p^{4} T^{9} + p^{5} T^{10} \)
83$C_2 \wr S_5$ \( 1 - 4 T + 270 T^{2} - 699 T^{3} + 37708 T^{4} - 83381 T^{5} + 37708 p T^{6} - 699 p^{2} T^{7} + 270 p^{3} T^{8} - 4 p^{4} T^{9} + p^{5} T^{10} \)
89$C_2 \wr S_5$ \( 1 + 3 T + 224 T^{2} + 1148 T^{3} + 33008 T^{4} + 116853 T^{5} + 33008 p T^{6} + 1148 p^{2} T^{7} + 224 p^{3} T^{8} + 3 p^{4} T^{9} + p^{5} T^{10} \)
97$C_2 \wr S_5$ \( 1 + 25 T + 342 T^{2} + 3288 T^{3} + 34066 T^{4} + 336739 T^{5} + 34066 p T^{6} + 3288 p^{2} T^{7} + 342 p^{3} T^{8} + 25 p^{4} T^{9} + p^{5} T^{10} \)
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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

−6.73803002023152352594981845449, −6.52525457489425418142522264060, −6.45136232710592368889883233471, −6.30360095761504171834735993112, −6.19730101318170557682295448525, −6.05367167695803447657544143764, −5.68607750296658854464745122184, −5.43331805179682219880286011836, −5.42510254810447795726185271561, −5.15752059399503719145480290644, −4.67814359315403346608570533665, −4.56044095766676496142132257705, −4.54150210189823859099802798523, −4.26556015538163957034862653700, −3.74626106451345101645177675842, −3.59720404918672105320782946095, −3.39970284215423413799154690200, −3.31723831501173889834771958840, −2.80311000091434026030416816829, −2.66708122768358550046141201273, −2.59047915425360233582599871283, −2.11439437883208474264829679160, −2.07238254625535039515444698720, −1.34820977695547805735406548244, −1.33954425156375303252036179499, 0, 0, 0, 0, 0, 1.33954425156375303252036179499, 1.34820977695547805735406548244, 2.07238254625535039515444698720, 2.11439437883208474264829679160, 2.59047915425360233582599871283, 2.66708122768358550046141201273, 2.80311000091434026030416816829, 3.31723831501173889834771958840, 3.39970284215423413799154690200, 3.59720404918672105320782946095, 3.74626106451345101645177675842, 4.26556015538163957034862653700, 4.54150210189823859099802798523, 4.56044095766676496142132257705, 4.67814359315403346608570533665, 5.15752059399503719145480290644, 5.42510254810447795726185271561, 5.43331805179682219880286011836, 5.68607750296658854464745122184, 6.05367167695803447657544143764, 6.19730101318170557682295448525, 6.30360095761504171834735993112, 6.45136232710592368889883233471, 6.52525457489425418142522264060, 6.73803002023152352594981845449

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