↑

Properties

Label 2-115-23.22-c6-0-20
Degree $2$
Conductor $115$
Sign $0.382 - 0.923i$
Analytic cond. $26.4562$
Root an. cond. $5.14356$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 13.8·2-s − 44.5·3-s + 127.·4-s + 55.9i·5-s − 616.·6-s − 282. i·7-s + 873.·8-s + 1.25e3·9-s + 772. i·10-s + 956. i·11-s − 5.66e3·12-s + 1.16e3·13-s − 3.90e3i·14-s − 2.49e3i·15-s + 3.93e3·16-s + 6.89e3i·17-s + ⋯
L(s)  = 1  + 1.72·2-s − 1.65·3-s + 1.98·4-s + 0.447i·5-s − 2.85·6-s − 0.822i·7-s + 1.70·8-s + 1.72·9-s + 0.772i·10-s + 0.718i·11-s − 3.28·12-s + 0.529·13-s − 1.42i·14-s − 0.738i·15-s + 0.961·16-s + 1.40i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(26.4562\)
Root analytic conductor: \(5.14356\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 115,\ (\ :3),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.934844202\)
\(L(\frac12)\) \(\approx\) \(2.934844202\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 55.9iT \)
23 \( 1 + (-1.12e4 - 4.65e3i)T \)
good2 \( 1 - 13.8T + 64T^{2} \)
3 \( 1 + 44.5T + 729T^{2} \)
7 \( 1 + 282. iT - 1.17e5T^{2} \)
11 \( 1 - 956. iT - 1.77e6T^{2} \)
13 \( 1 - 1.16e3T + 4.82e6T^{2} \)
17 \( 1 - 6.89e3iT - 2.41e7T^{2} \)
19 \( 1 - 1.20e4iT - 4.70e7T^{2} \)
29 \( 1 + 4.19e3T + 5.94e8T^{2} \)
31 \( 1 - 1.99e4T + 8.87e8T^{2} \)
37 \( 1 - 1.85e4iT - 2.56e9T^{2} \)
41 \( 1 - 6.24e4T + 4.75e9T^{2} \)
43 \( 1 - 2.43e3iT - 6.32e9T^{2} \)
47 \( 1 + 2.37e4T + 1.07e10T^{2} \)
53 \( 1 + 1.17e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.35e5T + 4.21e10T^{2} \)
61 \( 1 - 3.91e5iT - 5.15e10T^{2} \)
67 \( 1 + 2.94e5iT - 9.04e10T^{2} \)
71 \( 1 + 6.32e5T + 1.28e11T^{2} \)
73 \( 1 - 6.10e5T + 1.51e11T^{2} \)
79 \( 1 + 8.58e5iT - 2.43e11T^{2} \)
83 \( 1 + 7.44e4iT - 3.26e11T^{2} \)
89 \( 1 - 8.07e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.87e5iT - 8.32e11T^{2} \)
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

−12.57009066917549861218092402370, −11.82564905032681769597926242993, −10.83071441510082469922781134152, −10.28072002992214320957401979672, −7.49232856631092421857187570733, −6.42361011164639818471223927047, −5.80781323473935681470433887303, −4.57533744183635899752866645115, −3.67467422577158625382746151289, −1.49073441329429409910640230671, 0.69744386355520471034734212927, 2.78898438838245541710151606084, 4.57993148133256488010044039775, 5.24902928229893844103005446786, 6.07631049483748375477240986457, 7.01012847878855592258560827273, 9.109475433500631489074422731843, 11.01737898264047411127566007268, 11.40616556277732671482545432594, 12.26703415464750782557431946326

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