Properties

Label 2-2-1.1-c71-0-1
Degree $2$
Conductor $2$
Sign $1$
Analytic cond. $63.8492$
Root an. cond. $7.99057$
Motivic weight $71$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.43e10·2-s − 1.08e17·3-s + 1.18e21·4-s + 7.90e22·5-s + 3.72e27·6-s + 6.08e29·7-s − 4.05e31·8-s + 4.24e33·9-s − 2.71e33·10-s + 1.54e37·11-s − 1.27e38·12-s + 2.44e39·13-s − 2.09e40·14-s − 8.57e39·15-s + 1.39e42·16-s + 4.66e43·17-s − 1.45e44·18-s − 2.14e44·19-s + 9.33e43·20-s − 6.60e46·21-s − 5.32e47·22-s − 3.03e48·23-s + 4.39e48·24-s − 4.23e49·25-s − 8.39e49·26-s + 3.53e50·27-s + 7.18e50·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.25·3-s + 0.5·4-s + 0.0121·5-s + 0.884·6-s + 0.607·7-s − 0.353·8-s + 0.565·9-s − 0.00858·10-s + 1.66·11-s − 0.625·12-s + 0.696·13-s − 0.429·14-s − 0.0151·15-s + 0.250·16-s + 0.972·17-s − 0.399·18-s − 0.0862·19-s + 0.00607·20-s − 0.760·21-s − 1.17·22-s − 1.38·23-s + 0.442·24-s − 0.999·25-s − 0.492·26-s + 0.543·27-s + 0.303·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(72-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+71/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2\)
Sign: $1$
Analytic conductor: \(63.8492\)
Root analytic conductor: \(7.99057\)
Motivic weight: \(71\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2,\ (\ :71/2),\ 1)\)

Particular Values

\(L(36)\) \(\approx\) \(1.101551169\)
\(L(\frac12)\) \(\approx\) \(1.101551169\)
\(L(\frac{73}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 3.43e10T \)
good3 \( 1 + 1.08e17T + 7.50e33T^{2} \)
5 \( 1 - 7.90e22T + 4.23e49T^{2} \)
7 \( 1 - 6.08e29T + 1.00e60T^{2} \)
11 \( 1 - 1.54e37T + 8.68e73T^{2} \)
13 \( 1 - 2.44e39T + 1.23e79T^{2} \)
17 \( 1 - 4.66e43T + 2.30e87T^{2} \)
19 \( 1 + 2.14e44T + 6.18e90T^{2} \)
23 \( 1 + 3.03e48T + 4.81e96T^{2} \)
29 \( 1 - 5.23e51T + 6.76e103T^{2} \)
31 \( 1 + 2.94e52T + 7.70e105T^{2} \)
37 \( 1 + 4.30e54T + 2.19e111T^{2} \)
41 \( 1 + 5.89e56T + 3.21e114T^{2} \)
43 \( 1 - 6.09e57T + 9.46e115T^{2} \)
47 \( 1 + 3.56e59T + 5.23e118T^{2} \)
53 \( 1 - 3.09e61T + 2.65e122T^{2} \)
59 \( 1 - 1.36e63T + 5.37e125T^{2} \)
61 \( 1 + 1.19e63T + 5.73e126T^{2} \)
67 \( 1 - 1.19e65T + 4.48e129T^{2} \)
71 \( 1 + 3.22e65T + 2.75e131T^{2} \)
73 \( 1 - 2.31e66T + 1.97e132T^{2} \)
79 \( 1 + 9.14e66T + 5.38e134T^{2} \)
83 \( 1 + 1.67e68T + 1.79e136T^{2} \)
89 \( 1 + 7.15e68T + 2.55e138T^{2} \)
97 \( 1 + 2.08e70T + 1.15e141T^{2} \)
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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

−14.34165182299648934879441212027, −12.02150775375769971640294475246, −11.35044125004675883212595408967, −9.922385362819168541040472256629, −8.309308116085308896947760536579, −6.63912488618128702132546548529, −5.62805751643630772634374995411, −3.91079472184107988548123664514, −1.66064879833718373124201852285, −0.70844749435404216599064018316, 0.70844749435404216599064018316, 1.66064879833718373124201852285, 3.91079472184107988548123664514, 5.62805751643630772634374995411, 6.63912488618128702132546548529, 8.309308116085308896947760536579, 9.922385362819168541040472256629, 11.35044125004675883212595408967, 12.02150775375769971640294475246, 14.34165182299648934879441212027

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