↑

Properties

Label 2-115-23.22-c6-0-15
Degree $2$
Conductor $115$
Sign $-0.0388 - 0.999i$
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  − 11.9·2-s − 28.4·3-s + 77.7·4-s + 55.9i·5-s + 338.·6-s + 365. i·7-s − 163.·8-s + 77.6·9-s − 665. i·10-s + 1.71e3i·11-s − 2.20e3·12-s + 3.28e3·13-s − 4.35e3i·14-s − 1.58e3i·15-s − 3.02e3·16-s − 3.38e3i·17-s + ⋯
L(s)  = 1  − 1.48·2-s − 1.05·3-s + 1.21·4-s + 0.447i·5-s + 1.56·6-s + 1.06i·7-s − 0.319·8-s + 0.106·9-s − 0.665i·10-s + 1.28i·11-s − 1.27·12-s + 1.49·13-s − 1.58i·14-s − 0.470i·15-s − 0.739·16-s − 0.689i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0388 - 0.999i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0388 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.0388 - 0.999i$
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: \((2,\ 115,\ (\ :3),\ -0.0388 - 0.999i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.5654673041\)
\(L(\frac12)\) \(\approx\) \(0.5654673041\)
\(L(4)\) 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 \( 1 + (-1.21e4 + 472. i)T \)
good2 \( 1 + 11.9T + 64T^{2} \)
3 \( 1 + 28.4T + 729T^{2} \)
7 \( 1 - 365. iT - 1.17e5T^{2} \)
11 \( 1 - 1.71e3iT - 1.77e6T^{2} \)
13 \( 1 - 3.28e3T + 4.82e6T^{2} \)
17 \( 1 + 3.38e3iT - 2.41e7T^{2} \)
19 \( 1 + 8.53e3iT - 4.70e7T^{2} \)
29 \( 1 + 1.02e4T + 5.94e8T^{2} \)
31 \( 1 - 2.69e4T + 8.87e8T^{2} \)
37 \( 1 - 3.02e4iT - 2.56e9T^{2} \)
41 \( 1 + 2.24e4T + 4.75e9T^{2} \)
43 \( 1 + 6.30e3iT - 6.32e9T^{2} \)
47 \( 1 - 1.30e5T + 1.07e10T^{2} \)
53 \( 1 - 2.66e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.13e5T + 4.21e10T^{2} \)
61 \( 1 + 2.94e4iT - 5.15e10T^{2} \)
67 \( 1 + 3.26e5iT - 9.04e10T^{2} \)
71 \( 1 - 2.02e4T + 1.28e11T^{2} \)
73 \( 1 - 5.33e5T + 1.51e11T^{2} \)
79 \( 1 + 3.36e3iT - 2.43e11T^{2} \)
83 \( 1 + 8.04e5iT - 3.26e11T^{2} \)
89 \( 1 - 1.39e6iT - 4.96e11T^{2} \)
97 \( 1 + 7.19e5iT - 8.32e11T^{2} \)
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

−12.14369884209337528681880128835, −11.29921005785949775543123821812, −10.63441803723706093699470492187, −9.405953320717343832245141026439, −8.637330634280578417173006294468, −7.20258718098614133525540598296, −6.28840632605136819735481382339, −4.91573345320212651488935214703, −2.50378900751173154182423276110, −0.917397067945163860217263136927, 0.56272111856718680907554365609, 1.20913571088426316474375981433, 3.83722018482061920000773827285, 5.65323838055384614971272480788, 6.66990619885075312292294813218, 8.106916449384749218144006370228, 8.767168871186006690560430599998, 10.24601220896264950080998239956, 10.88689234115732184937567093697, 11.53707375990318358212520696746

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