Properties

Label 2-1-1.1-c31-0-1
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $6.08771$
Root an. cond. $2.46732$
Motivic weight $31$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.13e4·2-s + 3.08e7·3-s + 2.93e9·4-s − 8.25e10·5-s + 2.20e12·6-s + 1.14e13·7-s + 5.63e13·8-s + 3.34e14·9-s − 5.88e15·10-s − 2.40e15·11-s + 9.06e16·12-s − 2.01e17·13-s + 8.15e17·14-s − 2.54e18·15-s − 2.29e18·16-s + 1.09e19·17-s + 2.38e19·18-s − 1.42e19·19-s − 2.42e20·20-s + 3.52e20·21-s − 1.71e20·22-s − 3.85e18·23-s + 1.73e21·24-s + 2.15e21·25-s − 1.43e22·26-s − 8.74e21·27-s + 3.35e22·28-s + ⋯
L(s)  = 1  + 1.53·2-s + 1.24·3-s + 1.36·4-s − 1.20·5-s + 1.91·6-s + 0.910·7-s + 0.565·8-s + 0.541·9-s − 1.86·10-s − 0.173·11-s + 1.69·12-s − 1.09·13-s + 1.40·14-s − 1.50·15-s − 0.496·16-s + 0.925·17-s + 0.832·18-s − 0.215·19-s − 1.65·20-s + 1.13·21-s − 0.266·22-s − 0.00301·23-s + 0.702·24-s + 0.461·25-s − 1.68·26-s − 0.569·27-s + 1.24·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(32-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+31/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(6.08771\)
Root analytic conductor: \(2.46732\)
Motivic weight: \(31\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :31/2),\ 1)\)

Particular Values

\(L(16)\) \(\approx\) \(4.090583564\)
\(L(\frac12)\) \(\approx\) \(4.090583564\)
\(L(\frac{33}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 7.13e4T + 2.14e9T^{2} \)
3 \( 1 - 3.08e7T + 6.17e14T^{2} \)
5 \( 1 + 8.25e10T + 4.65e21T^{2} \)
7 \( 1 - 1.14e13T + 1.57e26T^{2} \)
11 \( 1 + 2.40e15T + 1.91e32T^{2} \)
13 \( 1 + 2.01e17T + 3.40e34T^{2} \)
17 \( 1 - 1.09e19T + 1.39e38T^{2} \)
19 \( 1 + 1.42e19T + 4.37e39T^{2} \)
23 \( 1 + 3.85e18T + 1.63e42T^{2} \)
29 \( 1 - 7.63e22T + 2.15e45T^{2} \)
31 \( 1 - 1.86e23T + 1.70e46T^{2} \)
37 \( 1 - 1.23e24T + 4.11e48T^{2} \)
41 \( 1 - 1.38e25T + 9.91e49T^{2} \)
43 \( 1 + 2.67e25T + 4.34e50T^{2} \)
47 \( 1 - 7.40e25T + 6.83e51T^{2} \)
53 \( 1 - 3.56e25T + 2.83e53T^{2} \)
59 \( 1 + 2.36e27T + 7.87e54T^{2} \)
61 \( 1 + 5.44e27T + 2.21e55T^{2} \)
67 \( 1 + 9.41e27T + 4.05e56T^{2} \)
71 \( 1 + 2.10e28T + 2.44e57T^{2} \)
73 \( 1 - 3.92e28T + 5.79e57T^{2} \)
79 \( 1 - 1.79e29T + 6.70e58T^{2} \)
83 \( 1 + 4.54e29T + 3.10e59T^{2} \)
89 \( 1 - 2.60e29T + 2.69e60T^{2} \)
97 \( 1 + 5.38e30T + 3.88e61T^{2} \)
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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

−24.74601604359586371830940866206, −23.33111690817342398940431907316, −21.15899146928555972189445545945, −19.67496808881181391172802518323, −15.25153064420284295903901090958, −14.18215877875086330379784124974, −12.00980514724768876514313786268, −7.899175154495250600673870691710, −4.48743998605134772879277686601, −2.85476292460552992262642155636, 2.85476292460552992262642155636, 4.48743998605134772879277686601, 7.899175154495250600673870691710, 12.00980514724768876514313786268, 14.18215877875086330379784124974, 15.25153064420284295903901090958, 19.67496808881181391172802518323, 21.15899146928555972189445545945, 23.33111690817342398940431907316, 24.74601604359586371830940866206

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