Properties

Label 2-1-1.1-c29-0-1
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $5.32780$
Root an. cond. $2.30820$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.60e4·2-s − 1.44e7·3-s + 1.42e8·4-s − 6.21e9·5-s − 3.77e11·6-s − 3.40e10·7-s − 1.02e13·8-s + 1.41e14·9-s − 1.62e14·10-s − 7.97e14·11-s − 2.07e15·12-s + 1.10e16·13-s − 8.87e14·14-s + 9.00e16·15-s − 3.44e17·16-s − 7.47e17·17-s + 3.68e18·18-s − 2.49e18·19-s − 8.88e17·20-s + 4.93e17·21-s − 2.07e19·22-s − 1.63e18·23-s + 1.48e20·24-s − 1.47e20·25-s + 2.88e20·26-s − 1.05e21·27-s − 4.86e18·28-s + ⋯
L(s)  = 1  + 1.12·2-s − 1.74·3-s + 0.266·4-s − 0.455·5-s − 1.96·6-s − 0.0189·7-s − 0.825·8-s + 2.06·9-s − 0.512·10-s − 0.633·11-s − 0.465·12-s + 0.778·13-s − 0.0213·14-s + 0.796·15-s − 1.19·16-s − 1.07·17-s + 2.31·18-s − 0.715·19-s − 0.121·20-s + 0.0331·21-s − 0.712·22-s − 0.0294·23-s + 1.44·24-s − 0.792·25-s + 0.875·26-s − 1.85·27-s − 0.00505·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(30-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+29/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(5.32780\)
Root analytic conductor: \(2.30820\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :29/2),\ -1)\)

Particular Values

\(L(15)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2.60e4T + 5.36e8T^{2} \)
3 \( 1 + 1.44e7T + 6.86e13T^{2} \)
5 \( 1 + 6.21e9T + 1.86e20T^{2} \)
7 \( 1 + 3.40e10T + 3.21e24T^{2} \)
11 \( 1 + 7.97e14T + 1.58e30T^{2} \)
13 \( 1 - 1.10e16T + 2.01e32T^{2} \)
17 \( 1 + 7.47e17T + 4.81e35T^{2} \)
19 \( 1 + 2.49e18T + 1.21e37T^{2} \)
23 \( 1 + 1.63e18T + 3.09e39T^{2} \)
29 \( 1 - 1.87e21T + 2.56e42T^{2} \)
31 \( 1 + 4.79e21T + 1.77e43T^{2} \)
37 \( 1 - 7.91e22T + 3.00e45T^{2} \)
41 \( 1 + 6.06e21T + 5.89e46T^{2} \)
43 \( 1 - 6.72e23T + 2.34e47T^{2} \)
47 \( 1 + 1.90e24T + 3.09e48T^{2} \)
53 \( 1 + 1.65e24T + 1.00e50T^{2} \)
59 \( 1 + 8.93e25T + 2.26e51T^{2} \)
61 \( 1 - 4.59e25T + 5.95e51T^{2} \)
67 \( 1 + 1.04e26T + 9.04e52T^{2} \)
71 \( 1 - 1.75e26T + 4.85e53T^{2} \)
73 \( 1 + 4.56e26T + 1.08e54T^{2} \)
79 \( 1 + 3.28e27T + 1.07e55T^{2} \)
83 \( 1 + 4.40e27T + 4.50e55T^{2} \)
89 \( 1 + 6.06e26T + 3.40e56T^{2} \)
97 \( 1 - 1.44e28T + 4.13e57T^{2} \)
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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

−23.85023141341665308092367887910, −22.99661949629539216345037865398, −21.58323327988740077110649105200, −17.98606264450376356129990123687, −15.79572792346414260372975867814, −12.86039942878683504422430358663, −11.22506451962109841988748838793, −6.12312213314869983040118889323, −4.46985241121272394578930720162, 0, 4.46985241121272394578930720162, 6.12312213314869983040118889323, 11.22506451962109841988748838793, 12.86039942878683504422430358663, 15.79572792346414260372975867814, 17.98606264450376356129990123687, 21.58323327988740077110649105200, 22.99661949629539216345037865398, 23.85023141341665308092367887910

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