Properties

Label 2-115-115.22-c3-0-4
Degree $2$
Conductor $115$
Sign $0.0990 - 0.995i$
Analytic cond. $6.78521$
Root an. cond. $2.60484$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 − 1.32i)2-s + (2.49 − 2.49i)3-s − 4.48i·4-s + (−3.09 + 10.7i)5-s − 6.60·6-s + (−19.9 + 19.9i)7-s + (−16.5 + 16.5i)8-s + 14.5i·9-s + (18.3 − 10.1i)10-s − 4.60i·11-s + (−11.1 − 11.1i)12-s + (8.15 − 8.15i)13-s + 52.8·14-s + (19.0 + 34.4i)15-s + 8.04·16-s + (−2.30 + 2.30i)17-s + ⋯
L(s)  = 1  + (−0.468 − 0.468i)2-s + (0.479 − 0.479i)3-s − 0.560i·4-s + (−0.276 + 0.961i)5-s − 0.449·6-s + (−1.07 + 1.07i)7-s + (−0.731 + 0.731i)8-s + 0.540i·9-s + (0.580 − 0.321i)10-s − 0.126i·11-s + (−0.268 − 0.268i)12-s + (0.173 − 0.173i)13-s + 1.00·14-s + (0.328 + 0.593i)15-s + 0.125·16-s + (−0.0328 + 0.0328i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0990 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0990 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.0990 - 0.995i$
Analytic conductor: \(6.78521\)
Root analytic conductor: \(2.60484\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3/2),\ 0.0990 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.463711 + 0.419861i\)
\(L(\frac12)\) \(\approx\) \(0.463711 + 0.419861i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.09 - 10.7i)T \)
23 \( 1 + (40.1 - 102. i)T \)
good2 \( 1 + (1.32 + 1.32i)T + 8iT^{2} \)
3 \( 1 + (-2.49 + 2.49i)T - 27iT^{2} \)
7 \( 1 + (19.9 - 19.9i)T - 343iT^{2} \)
11 \( 1 + 4.60iT - 1.33e3T^{2} \)
13 \( 1 + (-8.15 + 8.15i)T - 2.19e3iT^{2} \)
17 \( 1 + (2.30 - 2.30i)T - 4.91e3iT^{2} \)
19 \( 1 + 21.1T + 6.85e3T^{2} \)
29 \( 1 + 71.7iT - 2.43e4T^{2} \)
31 \( 1 + 80.9T + 2.97e4T^{2} \)
37 \( 1 + (135. - 135. i)T - 5.06e4iT^{2} \)
41 \( 1 + 72.5T + 6.89e4T^{2} \)
43 \( 1 + (137. + 137. i)T + 7.95e4iT^{2} \)
47 \( 1 + (-423. - 423. i)T + 1.03e5iT^{2} \)
53 \( 1 + (400. + 400. i)T + 1.48e5iT^{2} \)
59 \( 1 + 553. iT - 2.05e5T^{2} \)
61 \( 1 - 257. iT - 2.26e5T^{2} \)
67 \( 1 + (408. - 408. i)T - 3.00e5iT^{2} \)
71 \( 1 - 494.T + 3.57e5T^{2} \)
73 \( 1 + (396. - 396. i)T - 3.89e5iT^{2} \)
79 \( 1 - 100.T + 4.93e5T^{2} \)
83 \( 1 + (-461. - 461. i)T + 5.71e5iT^{2} \)
89 \( 1 - 1.01e3T + 7.04e5T^{2} \)
97 \( 1 + (929. - 929. i)T - 9.12e5iT^{2} \)
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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

−13.36154261915939228812717016149, −12.15239287820716106328165127850, −11.13213786536522774180554049560, −10.15172743856612737791292832385, −9.192195603880930103749934887009, −8.068485547901468933846971172233, −6.66135951212981168035913950315, −5.61944560048241821658634323352, −3.15003832510513934202611353112, −2.10931574829918353948315059177, 0.35265659985489280755289473136, 3.42468571102520068120232116505, 4.26847547465963520164802972692, 6.39757567085134343215819056024, 7.46146844173944725056702724684, 8.686849090437544772221981116017, 9.317528089952526317909127076443, 10.36461037423015515222980355337, 12.10915689449500847031569269463, 12.80115670644481919689787101409

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