Properties

Label 2-1-1.1-c31-0-0
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  − 3.13e4·2-s − 1.34e7·3-s − 1.16e9·4-s + 6.31e10·5-s + 4.22e11·6-s + 1.88e13·7-s + 1.03e14·8-s − 4.35e14·9-s − 1.97e15·10-s − 5.37e15·11-s + 1.57e16·12-s + 2.76e17·13-s − 5.89e17·14-s − 8.51e17·15-s − 7.53e17·16-s + 6.29e18·17-s + 1.36e19·18-s + 1.91e18·19-s − 7.35e19·20-s − 2.53e20·21-s + 1.68e20·22-s + 1.90e21·23-s − 1.40e21·24-s − 6.72e20·25-s − 8.67e21·26-s + 1.42e22·27-s − 2.19e22·28-s + ⋯
L(s)  = 1  − 0.676·2-s − 0.542·3-s − 0.542·4-s + 0.924·5-s + 0.367·6-s + 1.49·7-s + 1.04·8-s − 0.705·9-s − 0.625·10-s − 0.388·11-s + 0.294·12-s + 1.49·13-s − 1.01·14-s − 0.502·15-s − 0.163·16-s + 0.533·17-s + 0.477·18-s + 0.0289·19-s − 0.501·20-s − 0.813·21-s + 0.262·22-s + 1.48·23-s − 0.566·24-s − 0.144·25-s − 1.01·26-s + 0.925·27-s − 0.812·28-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

\(L(16)\) \(\approx\) \(1.080922784\)
\(L(\frac12)\) \(\approx\) \(1.080922784\)
\(L(\frac{33}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 3.13e4T + 2.14e9T^{2} \)
3 \( 1 + 1.34e7T + 6.17e14T^{2} \)
5 \( 1 - 6.31e10T + 4.65e21T^{2} \)
7 \( 1 - 1.88e13T + 1.57e26T^{2} \)
11 \( 1 + 5.37e15T + 1.91e32T^{2} \)
13 \( 1 - 2.76e17T + 3.40e34T^{2} \)
17 \( 1 - 6.29e18T + 1.39e38T^{2} \)
19 \( 1 - 1.91e18T + 4.37e39T^{2} \)
23 \( 1 - 1.90e21T + 1.63e42T^{2} \)
29 \( 1 - 5.22e22T + 2.15e45T^{2} \)
31 \( 1 + 6.09e22T + 1.70e46T^{2} \)
37 \( 1 + 2.07e24T + 4.11e48T^{2} \)
41 \( 1 + 5.09e24T + 9.91e49T^{2} \)
43 \( 1 - 8.39e24T + 4.34e50T^{2} \)
47 \( 1 - 2.13e25T + 6.83e51T^{2} \)
53 \( 1 - 1.59e26T + 2.83e53T^{2} \)
59 \( 1 - 2.16e27T + 7.87e54T^{2} \)
61 \( 1 + 6.60e27T + 2.21e55T^{2} \)
67 \( 1 + 2.77e26T + 4.05e56T^{2} \)
71 \( 1 - 7.68e28T + 2.44e57T^{2} \)
73 \( 1 - 2.29e28T + 5.79e57T^{2} \)
79 \( 1 + 2.99e29T + 6.70e58T^{2} \)
83 \( 1 - 1.89e29T + 3.10e59T^{2} \)
89 \( 1 + 2.41e30T + 2.69e60T^{2} \)
97 \( 1 + 3.68e30T + 3.88e61T^{2} \)
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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

−25.59815045482878068994052177070, −23.23077440410111646454398569488, −21.11130247220277519068551323863, −18.21187401009027052607053012703, −17.21085878815702513924416483796, −13.90495256268892525256775977574, −10.84651953735742116019517462645, −8.539617187885902554201134298215, −5.28996082164204900634330635759, −1.22818716050636831349375206415, 1.22818716050636831349375206415, 5.28996082164204900634330635759, 8.539617187885902554201134298215, 10.84651953735742116019517462645, 13.90495256268892525256775977574, 17.21085878815702513924416483796, 18.21187401009027052607053012703, 21.11130247220277519068551323863, 23.23077440410111646454398569488, 25.59815045482878068994052177070

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