Properties

Label 2-1-1.1-c51-0-3
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $16.4731$
Root an. cond. $4.05871$
Motivic weight $51$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.13e7·2-s + 2.68e12·3-s + 3.83e14·4-s + 4.84e17·5-s + 1.37e20·6-s + 2.59e21·7-s − 9.58e22·8-s + 5.03e24·9-s + 2.48e25·10-s − 4.45e26·11-s + 1.02e27·12-s − 1.37e28·13-s + 1.32e29·14-s + 1.29e30·15-s − 5.78e30·16-s + 2.62e31·17-s + 2.58e32·18-s + 1.11e32·19-s + 1.86e32·20-s + 6.94e33·21-s − 2.28e34·22-s + 2.36e34·23-s − 2.57e35·24-s − 2.09e35·25-s − 7.03e35·26-s + 7.71e36·27-s + 9.94e35·28-s + ⋯
L(s)  = 1  + 1.08·2-s + 1.82·3-s + 0.170·4-s + 0.727·5-s + 1.97·6-s + 0.729·7-s − 0.897·8-s + 2.33·9-s + 0.786·10-s − 1.23·11-s + 0.311·12-s − 0.538·13-s + 0.789·14-s + 1.32·15-s − 1.14·16-s + 1.10·17-s + 2.52·18-s + 0.273·19-s + 0.123·20-s + 1.33·21-s − 1.33·22-s + 0.445·23-s − 1.63·24-s − 0.471·25-s − 0.583·26-s + 2.44·27-s + 0.124·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(52-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+51/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(16.4731\)
Root analytic conductor: \(4.05871\)
Motivic weight: \(51\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :51/2),\ 1)\)

Particular Values

\(L(26)\) \(\approx\) \(5.711767548\)
\(L(\frac12)\) \(\approx\) \(5.711767548\)
\(L(\frac{53}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 5.13e7T + 2.25e15T^{2} \)
3 \( 1 - 2.68e12T + 2.15e24T^{2} \)
5 \( 1 - 4.84e17T + 4.44e35T^{2} \)
7 \( 1 - 2.59e21T + 1.25e43T^{2} \)
11 \( 1 + 4.45e26T + 1.29e53T^{2} \)
13 \( 1 + 1.37e28T + 6.47e56T^{2} \)
17 \( 1 - 2.62e31T + 5.66e62T^{2} \)
19 \( 1 - 1.11e32T + 1.64e65T^{2} \)
23 \( 1 - 2.36e34T + 2.80e69T^{2} \)
29 \( 1 + 2.25e37T + 3.82e74T^{2} \)
31 \( 1 + 4.11e37T + 1.14e76T^{2} \)
37 \( 1 - 5.65e39T + 9.51e79T^{2} \)
41 \( 1 + 5.54e40T + 1.78e82T^{2} \)
43 \( 1 + 5.70e41T + 2.02e83T^{2} \)
47 \( 1 - 1.49e42T + 1.89e85T^{2} \)
53 \( 1 + 1.42e44T + 8.67e87T^{2} \)
59 \( 1 - 2.24e44T + 2.05e90T^{2} \)
61 \( 1 - 2.34e44T + 1.12e91T^{2} \)
67 \( 1 + 2.65e45T + 1.34e93T^{2} \)
71 \( 1 - 7.64e46T + 2.59e94T^{2} \)
73 \( 1 - 5.90e47T + 1.07e95T^{2} \)
79 \( 1 - 3.15e48T + 6.01e96T^{2} \)
83 \( 1 + 1.40e48T + 7.46e97T^{2} \)
89 \( 1 + 1.71e48T + 2.62e99T^{2} \)
97 \( 1 - 6.68e50T + 2.11e101T^{2} \)
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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

−20.65855459376361729746132410119, −18.46338132095781233343618547286, −15.07846003640372603587177249012, −14.06237214085392692773852525535, −12.97643889751393502709158085272, −9.603932279762246098714017371305, −7.88468642974278430361491106397, −5.08956176803864243723829146291, −3.31909674112389602885775829792, −2.06773246701006938715445410458, 2.06773246701006938715445410458, 3.31909674112389602885775829792, 5.08956176803864243723829146291, 7.88468642974278430361491106397, 9.603932279762246098714017371305, 12.97643889751393502709158085272, 14.06237214085392692773852525535, 15.07846003640372603587177249012, 18.46338132095781233343618547286, 20.65855459376361729746132410119

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