Properties

Label 2-1-1.1-c75-0-5
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

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 3.67e11·2-s − 8.83e17·3-s + 9.72e22·4-s + 2.43e26·5-s − 3.24e29·6-s + 1.21e31·7-s + 2.18e34·8-s + 1.71e35·9-s + 8.95e37·10-s − 5.65e38·11-s − 8.58e40·12-s + 5.30e40·13-s + 4.46e42·14-s − 2.15e44·15-s + 4.35e45·16-s + 1.14e45·17-s + 6.30e46·18-s + 1.24e48·19-s + 2.36e49·20-s − 1.07e49·21-s − 2.07e50·22-s + 8.02e50·23-s − 1.92e52·24-s + 3.28e52·25-s + 1.94e52·26-s + 3.85e53·27-s + 1.18e54·28-s + ⋯
L(s)  = 1  + 1.89·2-s − 1.13·3-s + 2.57·4-s + 1.49·5-s − 2.14·6-s + 0.247·7-s + 2.97·8-s + 0.281·9-s + 2.83·10-s − 0.501·11-s − 2.91·12-s + 0.0894·13-s + 0.467·14-s − 1.69·15-s + 3.05·16-s + 0.0828·17-s + 0.533·18-s + 1.38·19-s + 3.85·20-s − 0.280·21-s − 0.947·22-s + 0.691·23-s − 3.36·24-s + 1.24·25-s + 0.169·26-s + 0.812·27-s + 0.636·28-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

\(L(38)\) \(\approx\) \(6.294403841\)
\(L(\frac12)\) \(\approx\) \(6.294403841\)
\(L(\frac{77}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 3.67e11T + 3.77e22T^{2} \)
3 \( 1 + 8.83e17T + 6.08e35T^{2} \)
5 \( 1 - 2.43e26T + 2.64e52T^{2} \)
7 \( 1 - 1.21e31T + 2.41e63T^{2} \)
11 \( 1 + 5.65e38T + 1.27e78T^{2} \)
13 \( 1 - 5.30e40T + 3.51e83T^{2} \)
17 \( 1 - 1.14e45T + 1.92e92T^{2} \)
19 \( 1 - 1.24e48T + 8.06e95T^{2} \)
23 \( 1 - 8.02e50T + 1.34e102T^{2} \)
29 \( 1 - 8.53e54T + 4.78e109T^{2} \)
31 \( 1 + 6.18e55T + 7.11e111T^{2} \)
37 \( 1 + 5.98e58T + 4.12e117T^{2} \)
41 \( 1 + 5.32e60T + 9.09e120T^{2} \)
43 \( 1 - 2.59e61T + 3.23e122T^{2} \)
47 \( 1 + 5.76e62T + 2.55e125T^{2} \)
53 \( 1 - 5.92e63T + 2.09e129T^{2} \)
59 \( 1 - 2.82e66T + 6.51e132T^{2} \)
61 \( 1 - 5.43e66T + 7.93e133T^{2} \)
67 \( 1 + 2.61e68T + 9.02e136T^{2} \)
71 \( 1 - 1.78e67T + 6.98e138T^{2} \)
73 \( 1 + 1.09e70T + 5.61e139T^{2} \)
79 \( 1 + 1.49e71T + 2.09e142T^{2} \)
83 \( 1 - 3.39e71T + 8.52e143T^{2} \)
89 \( 1 - 2.45e72T + 1.60e146T^{2} \)
97 \( 1 - 1.36e74T + 1.01e149T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−16.16229169962515869019078375745, −14.27440203825751164593063510913, −13.15523880421666537234992680990, −11.75483711706298129143441610081, −10.42360721707807538206410562582, −6.74678126540167852088474308539, −5.58577562851286952894089240118, −5.03437170072694803087955500738, −2.91172135971785883778611554615, −1.45137740352345554509119398267, 1.45137740352345554509119398267, 2.91172135971785883778611554615, 5.03437170072694803087955500738, 5.58577562851286952894089240118, 6.74678126540167852088474308539, 10.42360721707807538206410562582, 11.75483711706298129143441610081, 13.15523880421666537234992680990, 14.27440203825751164593063510913, 16.16229169962515869019078375745

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