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Properties

Label 2-115-5.4-c3-0-32
Degree $2$
Conductor $115$
Sign $-0.240 - 0.970i$
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  − 4.93i·2-s − 9.92i·3-s − 16.3·4-s + (10.8 − 2.68i)5-s − 48.9·6-s + 14.6i·7-s + 41.3i·8-s − 71.4·9-s + (−13.2 − 53.5i)10-s + 49.6·11-s + 162. i·12-s − 59.3i·13-s + 72.2·14-s + (−26.6 − 107. i)15-s + 73.3·16-s + 4.26i·17-s + ⋯
L(s)  = 1  − 1.74i·2-s − 1.90i·3-s − 2.04·4-s + (0.970 − 0.240i)5-s − 3.33·6-s + 0.789i·7-s + 1.82i·8-s − 2.64·9-s + (−0.419 − 1.69i)10-s + 1.36·11-s + 3.91i·12-s − 1.26i·13-s + 1.37·14-s + (−0.458 − 1.85i)15-s + 1.14·16-s + 0.0609i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.240 - 0.970i$
Analytic conductor: \(6.78521\)
Root analytic conductor: \(2.60484\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3/2),\ -0.240 - 0.970i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.951637 + 1.21576i\)
\(L(\frac12)\) \(\approx\) \(0.951637 + 1.21576i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-10.8 + 2.68i)T \)
23 \( 1 + 23iT \)
good2 \( 1 + 4.93iT - 8T^{2} \)
3 \( 1 + 9.92iT - 27T^{2} \)
7 \( 1 - 14.6iT - 343T^{2} \)
11 \( 1 - 49.6T + 1.33e3T^{2} \)
13 \( 1 + 59.3iT - 2.19e3T^{2} \)
17 \( 1 - 4.26iT - 4.91e3T^{2} \)
19 \( 1 - 2.76T + 6.85e3T^{2} \)
29 \( 1 - 180.T + 2.43e4T^{2} \)
31 \( 1 + 131.T + 2.97e4T^{2} \)
37 \( 1 - 225. iT - 5.06e4T^{2} \)
41 \( 1 + 241.T + 6.89e4T^{2} \)
43 \( 1 + 202. iT - 7.95e4T^{2} \)
47 \( 1 + 194. iT - 1.03e5T^{2} \)
53 \( 1 - 295. iT - 1.48e5T^{2} \)
59 \( 1 - 380.T + 2.05e5T^{2} \)
61 \( 1 - 97.7T + 2.26e5T^{2} \)
67 \( 1 + 584. iT - 3.00e5T^{2} \)
71 \( 1 + 799.T + 3.57e5T^{2} \)
73 \( 1 - 227. iT - 3.89e5T^{2} \)
79 \( 1 + 59.6T + 4.93e5T^{2} \)
83 \( 1 + 132. iT - 5.71e5T^{2} \)
89 \( 1 - 1.15e3T + 7.04e5T^{2} \)
97 \( 1 - 1.56e3iT - 9.12e5T^{2} \)
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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

−12.21339500081368422278902936154, −11.84803421089793988045179599796, −10.48553927473940328296717438219, −9.127297845756391130927822116012, −8.398360309849589885811763145849, −6.58987463069333916729260476069, −5.43690173442325544181071566000, −2.98621935681872501960621808177, −1.91383637138711682142426145544, −0.905920691010995417089604959946, 3.87952489453586495675054069076, 4.76202558436174032927239976137, 5.96569375117387135722491367206, 6.91461811776363799593637466842, 8.759674779350022306009916551787, 9.338277853092492879960525597404, 10.16716159961823906596949944897, 11.42875128712091851403030969184, 13.66221289567400994688750070997, 14.41111377284435077723157726814

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