Properties

Label 2-1-1.1-c75-0-4
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $35.6228$
Root an. cond. $5.96848$
Motivic weight $75$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.20e11·2-s + 1.08e18·3-s + 1.09e22·4-s − 2.30e25·5-s + 2.40e29·6-s + 1.96e31·7-s − 5.92e33·8-s + 5.77e35·9-s − 5.09e36·10-s + 9.40e38·11-s + 1.19e40·12-s + 1.14e42·13-s + 4.32e42·14-s − 2.51e43·15-s − 1.72e45·16-s + 1.79e46·17-s + 1.27e47·18-s + 1.43e48·19-s − 2.52e47·20-s + 2.13e49·21-s + 2.07e50·22-s − 1.23e51·23-s − 6.44e51·24-s − 2.59e52·25-s + 2.52e53·26-s − 3.37e52·27-s + 2.14e53·28-s + ⋯
L(s)  = 1  + 1.13·2-s + 1.39·3-s + 0.290·4-s − 0.141·5-s + 1.58·6-s + 0.399·7-s − 0.806·8-s + 0.948·9-s − 0.161·10-s + 0.833·11-s + 0.405·12-s + 1.92·13-s + 0.453·14-s − 0.197·15-s − 1.20·16-s + 1.29·17-s + 1.07·18-s + 1.60·19-s − 0.0411·20-s + 0.557·21-s + 0.947·22-s − 1.06·23-s − 1.12·24-s − 0.979·25-s + 2.18·26-s − 0.0712·27-s + 0.115·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(76-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+75/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(35.6228\)
Root analytic conductor: \(5.96848\)
Motivic weight: \(75\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :75/2),\ 1)\)

Particular Values

\(L(38)\) \(\approx\) \(6.075204432\)
\(L(\frac12)\) \(\approx\) \(6.075204432\)
\(L(\frac{77}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2.20e11T + 3.77e22T^{2} \)
3 \( 1 - 1.08e18T + 6.08e35T^{2} \)
5 \( 1 + 2.30e25T + 2.64e52T^{2} \)
7 \( 1 - 1.96e31T + 2.41e63T^{2} \)
11 \( 1 - 9.40e38T + 1.27e78T^{2} \)
13 \( 1 - 1.14e42T + 3.51e83T^{2} \)
17 \( 1 - 1.79e46T + 1.92e92T^{2} \)
19 \( 1 - 1.43e48T + 8.06e95T^{2} \)
23 \( 1 + 1.23e51T + 1.34e102T^{2} \)
29 \( 1 + 2.48e54T + 4.78e109T^{2} \)
31 \( 1 - 1.95e55T + 7.11e111T^{2} \)
37 \( 1 + 2.65e58T + 4.12e117T^{2} \)
41 \( 1 - 3.61e59T + 9.09e120T^{2} \)
43 \( 1 - 8.98e59T + 3.23e122T^{2} \)
47 \( 1 - 2.48e62T + 2.55e125T^{2} \)
53 \( 1 - 3.59e64T + 2.09e129T^{2} \)
59 \( 1 + 4.00e66T + 6.51e132T^{2} \)
61 \( 1 + 1.59e67T + 7.93e133T^{2} \)
67 \( 1 - 1.61e68T + 9.02e136T^{2} \)
71 \( 1 - 2.63e69T + 6.98e138T^{2} \)
73 \( 1 + 2.52e69T + 5.61e139T^{2} \)
79 \( 1 + 9.29e70T + 2.09e142T^{2} \)
83 \( 1 + 7.84e71T + 8.52e143T^{2} \)
89 \( 1 - 2.05e73T + 1.60e146T^{2} \)
97 \( 1 + 4.10e74T + 1.01e149T^{2} \)
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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

−15.67286189928881532815788560204, −14.23435972480361898972743107505, −13.66522268166784773511616252889, −11.82830003468771494801268332058, −9.303343689316899396095369159019, −7.981619235511270078094357245344, −5.83359990030070811993798679782, −3.91373573657624779746893925842, −3.26020102931758188694564734759, −1.44169842225286580724395584792, 1.44169842225286580724395584792, 3.26020102931758188694564734759, 3.91373573657624779746893925842, 5.83359990030070811993798679782, 7.981619235511270078094357245344, 9.303343689316899396095369159019, 11.82830003468771494801268332058, 13.66522268166784773511616252889, 14.23435972480361898972743107505, 15.67286189928881532815788560204

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