↑

Properties

Label 2-315-7.4-c1-0-4
Degree $2$
Conductor $315$
Sign $-0.106 - 0.994i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.933 + 1.61i)2-s + (−0.741 + 1.28i)4-s + (−0.5 − 0.866i)5-s + (1.69 + 2.03i)7-s + 0.965·8-s + (0.933 − 1.61i)10-s + (−2.67 + 4.63i)11-s + 4.34·13-s + (−1.70 + 4.63i)14-s + (2.38 + 4.12i)16-s + (−0.933 + 1.61i)17-s + (−2.62 − 4.54i)19-s + 1.48·20-s − 9.98·22-s + (−3.45 − 5.97i)23-s + ⋯
L(s)  = 1  + (0.659 + 1.14i)2-s + (−0.370 + 0.642i)4-s + (−0.223 − 0.387i)5-s + (0.639 + 0.768i)7-s + 0.341·8-s + (0.295 − 0.511i)10-s + (−0.806 + 1.39i)11-s + 1.20·13-s + (−0.456 + 1.23i)14-s + (0.595 + 1.03i)16-s + (−0.226 + 0.391i)17-s + (−0.602 − 1.04i)19-s + 0.331·20-s − 2.12·22-s + (−0.719 − 1.24i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.106 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.106 - 0.994i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.106 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27688 + 1.42105i\)
\(L(\frac12)\) \(\approx\) \(1.27688 + 1.42105i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-1.69 - 2.03i)T \)
good2 \( 1 + (-0.933 - 1.61i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + (2.67 - 4.63i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.34T + 13T^{2} \)
17 \( 1 + (0.933 - 1.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.62 + 4.54i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.45 + 5.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.38T + 29T^{2} \)
31 \( 1 + (-4.71 + 8.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.08 + 1.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.314T + 41T^{2} \)
43 \( 1 + 7.56T + 43T^{2} \)
47 \( 1 + (4.21 + 7.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.86 + 3.23i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.639 - 1.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.39 - 2.41i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.04 - 1.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.81T + 71T^{2} \)
73 \( 1 + (-2.81 + 4.87i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.97 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 + (-1.45 - 2.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.91T + 97T^{2} \)
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

−12.17224063866108400210852770424, −11.05435587386196781181949845755, −10.05386239012668757255614466324, −8.540917644744947861496178339243, −8.127587968906272371396507079880, −6.89975334331442527396893177801, −6.00558540312991373584856903280, −4.90599709971202123609620825790, −4.29298150549208159896463594817, −2.13548841692004273749629679118, 1.39818632352173413279053578920, 3.09761170036044919860526810520, 3.84626568541919462234037005297, 5.06334244274788804255641820821, 6.32908400646749551618930281144, 7.78745004067779238280812389133, 8.433164156695085148485486938954, 10.17283651835680331193215533126, 10.73038439665212166810248352464, 11.35360530613554731921133717570

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