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Properties

Label 2-315-63.25-c1-0-23
Degree $2$
Conductor $315$
Sign $-0.283 + 0.958i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.390·2-s + (0.919 − 1.46i)3-s − 1.84·4-s + (−0.5 + 0.866i)5-s + (−0.359 + 0.573i)6-s + (2.26 − 1.37i)7-s + 1.50·8-s + (−1.30 − 2.69i)9-s + (0.195 − 0.338i)10-s + (−1.55 − 2.69i)11-s + (−1.69 + 2.71i)12-s + (−1.07 − 1.86i)13-s + (−0.884 + 0.537i)14-s + (0.811 + 1.53i)15-s + 3.10·16-s + (−0.0261 + 0.0453i)17-s + ⋯
L(s)  = 1  − 0.276·2-s + (0.530 − 0.847i)3-s − 0.923·4-s + (−0.223 + 0.387i)5-s + (−0.146 + 0.234i)6-s + (0.854 − 0.519i)7-s + 0.531·8-s + (−0.436 − 0.899i)9-s + (0.0618 − 0.107i)10-s + (−0.469 − 0.813i)11-s + (−0.490 + 0.782i)12-s + (−0.298 − 0.517i)13-s + (−0.236 + 0.143i)14-s + (0.209 + 0.395i)15-s + 0.776·16-s + (−0.00634 + 0.0109i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.283 + 0.958i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.283 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586166 - 0.784522i\)
\(L(\frac12)\) \(\approx\) \(0.586166 - 0.784522i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.919 + 1.46i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.26 + 1.37i)T \)
good2 \( 1 + 0.390T + 2T^{2} \)
11 \( 1 + (1.55 + 2.69i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.07 + 1.86i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.0261 - 0.0453i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.73 + 6.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0525 + 0.0909i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.27 - 3.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.44T + 31T^{2} \)
37 \( 1 + (0.298 + 0.517i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.88 - 8.45i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.29 + 5.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.27T + 47T^{2} \)
53 \( 1 + (2.39 - 4.13i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.25T + 59T^{2} \)
61 \( 1 - 6.68T + 61T^{2} \)
67 \( 1 - 2.84T + 67T^{2} \)
71 \( 1 - 6.22T + 71T^{2} \)
73 \( 1 + (2.38 - 4.13i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + (7.38 - 12.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.22 - 7.31i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.575 + 0.996i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.24984792796839767802932617096, −10.54081910294012421879856289128, −9.255759352356625043117368999598, −8.360933649405388300904658315358, −7.81950756632820419597589160623, −6.80137225785909644623915633682, −5.34785338923708636233102176858, −4.10184606338393955070724321465, −2.69406215662727957406106433095, −0.77748402934216667418461062215, 2.11179087002346787986386585242, 4.02833783439043280438169347524, 4.65943049518545570554196705414, 5.63345512345499123201096423754, 7.72083983170443507913416512142, 8.268651264115848465575822648810, 9.125909318347533890200198124510, 9.899746721337109068994091716646, 10.71865532776322807748026692535, 11.95809085556635099386834994146

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