Properties

Label 2-1-1.1-c75-0-0
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  + 9.60e10·2-s − 6.47e17·3-s − 2.85e22·4-s − 1.49e26·5-s − 6.21e28·6-s − 3.16e31·7-s − 6.37e33·8-s − 1.89e35·9-s − 1.43e37·10-s − 9.04e38·11-s + 1.84e40·12-s − 1.02e42·13-s − 3.03e42·14-s + 9.68e43·15-s + 4.66e44·16-s + 2.82e45·17-s − 1.81e46·18-s + 5.13e47·19-s + 4.27e48·20-s + 2.04e49·21-s − 8.68e49·22-s − 1.06e51·23-s + 4.12e51·24-s − 4.07e51·25-s − 9.83e52·26-s + 5.16e53·27-s + 9.02e53·28-s + ⋯
L(s)  = 1  + 0.494·2-s − 0.829·3-s − 0.755·4-s − 0.919·5-s − 0.410·6-s − 0.643·7-s − 0.867·8-s − 0.311·9-s − 0.454·10-s − 0.801·11-s + 0.627·12-s − 1.72·13-s − 0.318·14-s + 0.763·15-s + 0.326·16-s + 0.203·17-s − 0.153·18-s + 0.571·19-s + 0.695·20-s + 0.534·21-s − 0.396·22-s − 0.920·23-s + 0.720·24-s − 0.153·25-s − 0.853·26-s + 1.08·27-s + 0.486·28-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

\(L(38)\) \(\approx\) \(0.01821928660\)
\(L(\frac12)\) \(\approx\) \(0.01821928660\)
\(L(\frac{77}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 9.60e10T + 3.77e22T^{2} \)
3 \( 1 + 6.47e17T + 6.08e35T^{2} \)
5 \( 1 + 1.49e26T + 2.64e52T^{2} \)
7 \( 1 + 3.16e31T + 2.41e63T^{2} \)
11 \( 1 + 9.04e38T + 1.27e78T^{2} \)
13 \( 1 + 1.02e42T + 3.51e83T^{2} \)
17 \( 1 - 2.82e45T + 1.92e92T^{2} \)
19 \( 1 - 5.13e47T + 8.06e95T^{2} \)
23 \( 1 + 1.06e51T + 1.34e102T^{2} \)
29 \( 1 + 5.54e54T + 4.78e109T^{2} \)
31 \( 1 + 6.85e55T + 7.11e111T^{2} \)
37 \( 1 - 1.00e59T + 4.12e117T^{2} \)
41 \( 1 - 4.31e60T + 9.09e120T^{2} \)
43 \( 1 + 1.74e61T + 3.23e122T^{2} \)
47 \( 1 + 9.73e62T + 2.55e125T^{2} \)
53 \( 1 + 2.55e63T + 2.09e129T^{2} \)
59 \( 1 - 5.98e65T + 6.51e132T^{2} \)
61 \( 1 + 1.23e67T + 7.93e133T^{2} \)
67 \( 1 + 9.36e67T + 9.02e136T^{2} \)
71 \( 1 + 3.34e69T + 6.98e138T^{2} \)
73 \( 1 + 1.44e70T + 5.61e139T^{2} \)
79 \( 1 - 1.89e71T + 2.09e142T^{2} \)
83 \( 1 + 1.25e72T + 8.52e143T^{2} \)
89 \( 1 + 1.25e73T + 1.60e146T^{2} \)
97 \( 1 + 8.39e73T + 1.01e149T^{2} \)
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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

−16.41194922677465350423826685558, −14.70124889196705228139142433963, −12.80846359722859386103123185603, −11.69809995216971533333574557631, −9.740511050496170983606681222587, −7.73118488806740473889090445854, −5.75025107802660670760758764342, −4.57885962660034381012798481313, −3.03941673987890677993991111104, −0.07425016818954904161437391674, 0.07425016818954904161437391674, 3.03941673987890677993991111104, 4.57885962660034381012798481313, 5.75025107802660670760758764342, 7.73118488806740473889090445854, 9.740511050496170983606681222587, 11.69809995216971533333574557631, 12.80846359722859386103123185603, 14.70124889196705228139142433963, 16.41194922677465350423826685558

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