Properties

Label 2-10-1.1-c13-0-2
Degree $2$
Conductor $10$
Sign $-1$
Analytic cond. $10.7230$
Root an. cond. $3.27461$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 64·2-s − 26·3-s + 4.09e3·4-s − 1.56e4·5-s + 1.66e3·6-s + 5.38e5·7-s − 2.62e5·8-s − 1.59e6·9-s + 1.00e6·10-s − 3.99e6·11-s − 1.06e5·12-s − 2.38e7·13-s − 3.44e7·14-s + 4.06e5·15-s + 1.67e7·16-s − 1.92e8·17-s + 1.01e8·18-s + 1.66e8·19-s − 6.40e7·20-s − 1.40e7·21-s + 2.55e8·22-s − 3.66e8·23-s + 6.81e6·24-s + 2.44e8·25-s + 1.52e9·26-s + 8.28e7·27-s + 2.20e9·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.0205·3-s + 1/2·4-s − 0.447·5-s + 0.0145·6-s + 1.73·7-s − 0.353·8-s − 0.999·9-s + 0.316·10-s − 0.679·11-s − 0.0102·12-s − 1.36·13-s − 1.22·14-s + 0.00920·15-s + 1/4·16-s − 1.93·17-s + 0.706·18-s + 0.811·19-s − 0.223·20-s − 0.0356·21-s + 0.480·22-s − 0.516·23-s + 0.00728·24-s + 1/5·25-s + 0.968·26-s + 0.0411·27-s + 0.865·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-1$
Analytic conductor: \(10.7230\)
Root analytic conductor: \(3.27461\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10,\ (\ :13/2),\ -1)\)

Particular Values

\(L(7)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{6} T \)
5 \( 1 + p^{6} T \)
good3 \( 1 + 26 T + p^{13} T^{2} \)
7 \( 1 - 76934 p T + p^{13} T^{2} \)
11 \( 1 + 363168 p T + p^{13} T^{2} \)
13 \( 1 + 23834446 T + p^{13} T^{2} \)
17 \( 1 + 192273222 T + p^{13} T^{2} \)
19 \( 1 - 166485740 T + p^{13} T^{2} \)
23 \( 1 + 366866946 T + p^{13} T^{2} \)
29 \( 1 - 989855670 T + p^{13} T^{2} \)
31 \( 1 + 3445048468 T + p^{13} T^{2} \)
37 \( 1 + 29429851822 T + p^{13} T^{2} \)
41 \( 1 - 7043712582 T + p^{13} T^{2} \)
43 \( 1 - 8228005214 T + p^{13} T^{2} \)
47 \( 1 - 45741859938 T + p^{13} T^{2} \)
53 \( 1 + 90591954486 T + p^{13} T^{2} \)
59 \( 1 - 126033098940 T + p^{13} T^{2} \)
61 \( 1 + 292123673038 T + p^{13} T^{2} \)
67 \( 1 - 572402067098 T + p^{13} T^{2} \)
71 \( 1 + 1284329422908 T + p^{13} T^{2} \)
73 \( 1 - 196494986594 T + p^{13} T^{2} \)
79 \( 1 - 3776797097000 T + p^{13} T^{2} \)
83 \( 1 + 4556844205746 T + p^{13} T^{2} \)
89 \( 1 - 3748393684890 T + p^{13} T^{2} \)
97 \( 1 + 2743981383742 T + p^{13} T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.28236805244698199893653210928, −15.49537667826023387680416424718, −14.23811847267004920085271440081, −11.86099267144663710576334497312, −10.84110787001852964696272359890, −8.710718539615435959952989541319, −7.51821927702662037354979076128, −5.00782424948703379719266171494, −2.21969145174946915955485347354, 0, 2.21969145174946915955485347354, 5.00782424948703379719266171494, 7.51821927702662037354979076128, 8.710718539615435959952989541319, 10.84110787001852964696272359890, 11.86099267144663710576334497312, 14.23811847267004920085271440081, 15.49537667826023387680416424718, 17.28236805244698199893653210928

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