Properties

Label 2-1-1.1-c75-0-1
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  − 2.89e11·2-s + 1.03e18·3-s + 4.61e22·4-s − 2.47e26·5-s − 3.00e29·6-s − 6.83e31·7-s − 2.41e33·8-s + 4.64e35·9-s + 7.16e37·10-s − 1.33e39·11-s + 4.77e40·12-s + 2.92e41·13-s + 1.97e43·14-s − 2.56e44·15-s − 1.04e45·16-s + 4.46e44·17-s − 1.34e47·18-s − 8.32e47·19-s − 1.14e49·20-s − 7.07e49·21-s + 3.86e50·22-s + 1.13e51·23-s − 2.50e51·24-s + 3.47e52·25-s − 8.47e52·26-s − 1.48e53·27-s − 3.15e54·28-s + ⋯
L(s)  = 1  − 1.49·2-s + 1.32·3-s + 1.22·4-s − 1.52·5-s − 1.97·6-s − 1.39·7-s − 0.328·8-s + 0.763·9-s + 2.26·10-s − 1.18·11-s + 1.62·12-s + 0.493·13-s + 2.07·14-s − 2.02·15-s − 0.730·16-s + 0.0321·17-s − 1.13·18-s − 0.926·19-s − 1.85·20-s − 1.84·21-s + 1.76·22-s + 0.976·23-s − 0.436·24-s + 1.31·25-s − 0.735·26-s − 0.313·27-s − 1.69·28-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

\(L(38)\) \(\approx\) \(0.4817922437\)
\(L(\frac12)\) \(\approx\) \(0.4817922437\)
\(L(\frac{77}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 2.89e11T + 3.77e22T^{2} \)
3 \( 1 - 1.03e18T + 6.08e35T^{2} \)
5 \( 1 + 2.47e26T + 2.64e52T^{2} \)
7 \( 1 + 6.83e31T + 2.41e63T^{2} \)
11 \( 1 + 1.33e39T + 1.27e78T^{2} \)
13 \( 1 - 2.92e41T + 3.51e83T^{2} \)
17 \( 1 - 4.46e44T + 1.92e92T^{2} \)
19 \( 1 + 8.32e47T + 8.06e95T^{2} \)
23 \( 1 - 1.13e51T + 1.34e102T^{2} \)
29 \( 1 - 3.24e54T + 4.78e109T^{2} \)
31 \( 1 + 1.48e56T + 7.11e111T^{2} \)
37 \( 1 - 2.52e58T + 4.12e117T^{2} \)
41 \( 1 + 1.57e60T + 9.09e120T^{2} \)
43 \( 1 - 2.24e61T + 3.23e122T^{2} \)
47 \( 1 - 3.99e62T + 2.55e125T^{2} \)
53 \( 1 - 2.56e64T + 2.09e129T^{2} \)
59 \( 1 - 1.98e66T + 6.51e132T^{2} \)
61 \( 1 + 1.77e66T + 7.93e133T^{2} \)
67 \( 1 - 3.79e68T + 9.02e136T^{2} \)
71 \( 1 - 1.15e69T + 6.98e138T^{2} \)
73 \( 1 + 5.75e69T + 5.61e139T^{2} \)
79 \( 1 - 6.59e70T + 2.09e142T^{2} \)
83 \( 1 - 1.04e72T + 8.52e143T^{2} \)
89 \( 1 - 1.34e73T + 1.60e146T^{2} \)
97 \( 1 + 5.17e74T + 1.01e149T^{2} \)
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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

−16.27600116982798116923691995254, −15.33686387649280416526202783598, −12.99433846952762269445629494873, −10.73943178191518419056077062310, −9.166434742618446545633211301906, −8.189914496586127309609984085416, −7.18004539728659984538861799288, −3.72886977494453316318830446873, −2.55408352758006985195857895610, −0.46629182134089608875322244810, 0.46629182134089608875322244810, 2.55408352758006985195857895610, 3.72886977494453316318830446873, 7.18004539728659984538861799288, 8.189914496586127309609984085416, 9.166434742618446545633211301906, 10.73943178191518419056077062310, 12.99433846952762269445629494873, 15.33686387649280416526202783598, 16.27600116982798116923691995254

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