Properties

Label 2-1-1.1-c75-0-2
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  − 3.26e11·2-s − 1.51e18·3-s + 6.89e22·4-s − 4.33e25·5-s + 4.94e29·6-s + 4.24e31·7-s − 1.01e34·8-s + 1.68e36·9-s + 1.41e37·10-s + 1.56e38·11-s − 1.04e41·12-s − 1.88e41·13-s − 1.38e43·14-s + 6.56e43·15-s + 7.25e44·16-s + 9.86e45·17-s − 5.50e47·18-s − 1.40e47·19-s − 2.98e48·20-s − 6.43e49·21-s − 5.11e49·22-s + 8.97e50·23-s + 1.54e52·24-s − 2.45e52·25-s + 6.15e52·26-s − 1.63e54·27-s + 2.92e54·28-s + ⋯
L(s)  = 1  − 1.68·2-s − 1.94·3-s + 1.82·4-s − 0.266·5-s + 3.26·6-s + 0.864·7-s − 1.38·8-s + 2.77·9-s + 0.447·10-s + 0.138·11-s − 3.54·12-s − 0.317·13-s − 1.45·14-s + 0.517·15-s + 0.508·16-s + 0.711·17-s − 4.65·18-s − 0.156·19-s − 0.486·20-s − 1.67·21-s − 0.233·22-s + 0.772·23-s + 2.69·24-s − 0.929·25-s + 0.534·26-s − 3.43·27-s + 1.57·28-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

\(L(38)\) \(\approx\) \(0.3939146891\)
\(L(\frac12)\) \(\approx\) \(0.3939146891\)
\(L(\frac{77}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 3.26e11T + 3.77e22T^{2} \)
3 \( 1 + 1.51e18T + 6.08e35T^{2} \)
5 \( 1 + 4.33e25T + 2.64e52T^{2} \)
7 \( 1 - 4.24e31T + 2.41e63T^{2} \)
11 \( 1 - 1.56e38T + 1.27e78T^{2} \)
13 \( 1 + 1.88e41T + 3.51e83T^{2} \)
17 \( 1 - 9.86e45T + 1.92e92T^{2} \)
19 \( 1 + 1.40e47T + 8.06e95T^{2} \)
23 \( 1 - 8.97e50T + 1.34e102T^{2} \)
29 \( 1 - 7.48e54T + 4.78e109T^{2} \)
31 \( 1 - 9.83e55T + 7.11e111T^{2} \)
37 \( 1 + 4.42e58T + 4.12e117T^{2} \)
41 \( 1 - 3.18e60T + 9.09e120T^{2} \)
43 \( 1 + 2.58e61T + 3.23e122T^{2} \)
47 \( 1 + 4.72e61T + 2.55e125T^{2} \)
53 \( 1 + 7.75e64T + 2.09e129T^{2} \)
59 \( 1 + 8.17e64T + 6.51e132T^{2} \)
61 \( 1 + 6.09e66T + 7.93e133T^{2} \)
67 \( 1 - 4.31e68T + 9.02e136T^{2} \)
71 \( 1 - 2.46e69T + 6.98e138T^{2} \)
73 \( 1 + 4.09e69T + 5.61e139T^{2} \)
79 \( 1 - 9.26e69T + 2.09e142T^{2} \)
83 \( 1 + 2.78e69T + 8.52e143T^{2} \)
89 \( 1 - 1.96e73T + 1.60e146T^{2} \)
97 \( 1 - 2.01e74T + 1.01e149T^{2} \)
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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

−17.05176392275261344354531244194, −15.76235928230335292583310201977, −12.00951034391048467688486389958, −11.07180740004960860022466154762, −9.940437491072490005411445146049, −7.83905521870491387148149465878, −6.51922180212410194763033250794, −4.85290185031793539369583572704, −1.53357852812329393726544591288, −0.58393523381531499715284794994, 0.58393523381531499715284794994, 1.53357852812329393726544591288, 4.85290185031793539369583572704, 6.51922180212410194763033250794, 7.83905521870491387148149465878, 9.940437491072490005411445146049, 11.07180740004960860022466154762, 12.00951034391048467688486389958, 15.76235928230335292583310201977, 17.05176392275261344354531244194

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