Properties

Label 2-1-1.1-c63-0-3
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $25.1360$
Root an. cond. $5.01359$
Motivic weight $63$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.14e9·2-s + 2.06e15·3-s − 7.90e18·4-s − 9.56e21·5-s − 2.37e24·6-s − 1.15e26·7-s + 1.96e28·8-s + 3.13e30·9-s + 1.10e31·10-s − 2.43e32·11-s − 1.63e34·12-s + 1.25e35·13-s + 1.32e35·14-s − 1.98e37·15-s + 5.02e37·16-s + 7.86e38·17-s − 3.60e39·18-s − 2.70e39·19-s + 7.56e40·20-s − 2.38e41·21-s + 2.80e41·22-s + 3.63e42·23-s + 4.07e43·24-s − 1.68e43·25-s − 1.43e44·26-s + 4.12e45·27-s + 9.09e44·28-s + ⋯
L(s)  = 1  − 0.378·2-s + 1.93·3-s − 0.856·4-s − 0.919·5-s − 0.732·6-s − 0.275·7-s + 0.702·8-s + 2.74·9-s + 0.347·10-s − 0.383·11-s − 1.65·12-s + 1.01·13-s + 0.104·14-s − 1.77·15-s + 0.590·16-s + 1.36·17-s − 1.03·18-s − 0.141·19-s + 0.787·20-s − 0.533·21-s + 0.145·22-s + 0.463·23-s + 1.35·24-s − 0.155·25-s − 0.385·26-s + 3.36·27-s + 0.236·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(64-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+63/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(25.1360\)
Root analytic conductor: \(5.01359\)
Motivic weight: \(63\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :63/2),\ 1)\)

Particular Values

\(L(32)\) \(\approx\) \(2.400581862\)
\(L(\frac12)\) \(\approx\) \(2.400581862\)
\(L(\frac{65}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.14e9T + 9.22e18T^{2} \)
3 \( 1 - 2.06e15T + 1.14e30T^{2} \)
5 \( 1 + 9.56e21T + 1.08e44T^{2} \)
7 \( 1 + 1.15e26T + 1.74e53T^{2} \)
11 \( 1 + 2.43e32T + 4.05e65T^{2} \)
13 \( 1 - 1.25e35T + 1.50e70T^{2} \)
17 \( 1 - 7.86e38T + 3.29e77T^{2} \)
19 \( 1 + 2.70e39T + 3.64e80T^{2} \)
23 \( 1 - 3.63e42T + 6.14e85T^{2} \)
29 \( 1 - 3.93e45T + 1.35e92T^{2} \)
31 \( 1 - 1.18e47T + 9.03e93T^{2} \)
37 \( 1 - 1.80e49T + 6.26e98T^{2} \)
41 \( 1 - 3.23e50T + 4.03e101T^{2} \)
43 \( 1 - 8.59e50T + 8.10e102T^{2} \)
47 \( 1 - 1.26e52T + 2.19e105T^{2} \)
53 \( 1 + 5.19e53T + 4.25e108T^{2} \)
59 \( 1 + 4.37e55T + 3.66e111T^{2} \)
61 \( 1 - 2.79e56T + 2.99e112T^{2} \)
67 \( 1 - 1.12e57T + 1.10e115T^{2} \)
71 \( 1 - 1.56e58T + 4.25e116T^{2} \)
73 \( 1 + 9.21e58T + 2.45e117T^{2} \)
79 \( 1 + 4.18e59T + 3.55e119T^{2} \)
83 \( 1 + 3.28e60T + 7.97e120T^{2} \)
89 \( 1 + 3.02e61T + 6.47e122T^{2} \)
97 \( 1 - 5.45e61T + 1.46e125T^{2} \)
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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

−18.65327828711500423459760453036, −15.78724377848823859329320618624, −14.27896899969831979020939700695, −12.98942766431841316310760095896, −9.910172003100438307634546782256, −8.529931552949539470805252488505, −7.71513151385096268331106059192, −4.18209228791014334376063149687, −3.13322097782405085288527911067, −1.07806871839496369775651965234, 1.07806871839496369775651965234, 3.13322097782405085288527911067, 4.18209228791014334376063149687, 7.71513151385096268331106059192, 8.529931552949539470805252488505, 9.910172003100438307634546782256, 12.98942766431841316310760095896, 14.27896899969831979020939700695, 15.78724377848823859329320618624, 18.65327828711500423459760453036

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