↑

Properties

Label 2-315-63.58-c1-0-31
Degree $2$
Conductor $315$
Sign $-0.999 + 0.0238i$
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  + 0.0255·2-s + (0.676 − 1.59i)3-s − 1.99·4-s + (−0.5 − 0.866i)5-s + (0.0172 − 0.0407i)6-s + (−2.37 − 1.17i)7-s − 0.102·8-s + (−2.08 − 2.15i)9-s + (−0.0127 − 0.0221i)10-s + (−1.89 + 3.27i)11-s + (−1.35 + 3.18i)12-s + (−2.77 + 4.79i)13-s + (−0.0606 − 0.0300i)14-s + (−1.71 + 0.211i)15-s + 3.99·16-s + (−0.271 − 0.471i)17-s + ⋯
L(s)  = 1  + 0.0180·2-s + (0.390 − 0.920i)3-s − 0.999·4-s + (−0.223 − 0.387i)5-s + (0.00706 − 0.0166i)6-s + (−0.896 − 0.443i)7-s − 0.0361·8-s + (−0.695 − 0.718i)9-s + (−0.00404 − 0.00700i)10-s + (−0.569 + 0.987i)11-s + (−0.390 + 0.920i)12-s + (−0.768 + 1.33i)13-s + (−0.0162 − 0.00801i)14-s + (−0.443 + 0.0546i)15-s + 0.999·16-s + (−0.0659 − 0.114i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.999 + 0.0238i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.999 + 0.0238i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00552512 - 0.464048i\)
\(L(\frac12)\) \(\approx\) \(0.00552512 - 0.464048i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.676 + 1.59i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.37 + 1.17i)T \)
good2 \( 1 - 0.0255T + 2T^{2} \)
11 \( 1 + (1.89 - 3.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.77 - 4.79i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.271 + 0.471i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.62 + 6.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.62 + 6.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.04 + 7.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.74T + 31T^{2} \)
37 \( 1 + (-1.67 + 2.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.238 + 0.412i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.279 + 0.483i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 8.57T + 47T^{2} \)
53 \( 1 + (4.26 + 7.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 1.40T + 59T^{2} \)
61 \( 1 - 1.75T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 + (-2.91 - 5.04i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 2.96T + 79T^{2} \)
83 \( 1 + (3.97 + 6.88i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (9.02 - 15.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.81 + 6.60i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.47860580962422568585647505840, −9.731136431126561308303023624967, −9.484892008944898389958188810706, −8.317693937142392020938874692434, −7.35903551162794860722538133151, −6.54178565666525914953523476737, −4.98547761414982349492126769349, −3.99529557590330151846589310091, −2.40611234249644732157982137557, −0.31232629095532873996683633757, 3.12300600775413196373424045797, 3.59675588109139069135245816457, 5.24966525933220705235920394515, 5.78239678459674619877108612797, 7.77212476523389883389376957976, 8.354445852789044501978141988858, 9.611595499147578189476656809734, 9.940782739443216312350938163448, 10.93125567413833816446039226508, 12.21867614253505653917125187051

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