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Properties

Label 2-315-105.2-c1-0-12
Degree $2$
Conductor $315$
Sign $0.533 + 0.845i$
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  + (−0.0991 − 0.370i)2-s + (1.60 − 0.926i)4-s + (1.78 − 1.34i)5-s + (2.39 − 1.13i)7-s + (−1.04 − 1.04i)8-s + (−0.673 − 0.529i)10-s + (−3.48 + 2.01i)11-s + (−2.94 + 2.94i)13-s + (−0.656 − 0.772i)14-s + (1.57 − 2.71i)16-s + (−1.90 − 0.510i)17-s + (3.06 + 1.77i)19-s + (1.62 − 3.81i)20-s + (1.09 + 1.09i)22-s + (−1.74 + 0.467i)23-s + ⋯
L(s)  = 1  + (−0.0701 − 0.261i)2-s + (0.802 − 0.463i)4-s + (0.800 − 0.599i)5-s + (0.903 − 0.427i)7-s + (−0.369 − 0.369i)8-s + (−0.213 − 0.167i)10-s + (−1.05 + 0.606i)11-s + (−0.815 + 0.815i)13-s + (−0.175 − 0.206i)14-s + (0.392 − 0.679i)16-s + (−0.462 − 0.123i)17-s + (0.703 + 0.406i)19-s + (0.364 − 0.851i)20-s + (0.232 + 0.232i)22-s + (−0.363 + 0.0974i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.533 + 0.845i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47612 - 0.813772i\)
\(L(\frac12)\) \(\approx\) \(1.47612 - 0.813772i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.78 + 1.34i)T \)
7 \( 1 + (-2.39 + 1.13i)T \)
good2 \( 1 + (0.0991 + 0.370i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (3.48 - 2.01i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.94 - 2.94i)T - 13iT^{2} \)
17 \( 1 + (1.90 + 0.510i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.06 - 1.77i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.74 - 0.467i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 7.33T + 29T^{2} \)
31 \( 1 + (-1.20 - 2.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.29 - 2.22i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.20iT - 41T^{2} \)
43 \( 1 + (1.29 - 1.29i)T - 43iT^{2} \)
47 \( 1 + (-0.286 - 1.06i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.89 + 10.7i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-7.20 - 12.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.48 - 4.30i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.59 - 5.94i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.99iT - 71T^{2} \)
73 \( 1 + (12.5 + 3.35i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (12.9 + 7.47i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-11.2 - 11.2i)T + 83iT^{2} \)
89 \( 1 + (-4.22 + 7.31i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.46 - 4.46i)T + 97iT^{2} \)
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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

−11.61905310499701233226139539173, −10.22475967574674223060618937404, −10.12695469428107998878664338580, −8.775589630046916488850724889457, −7.59619363150255780200639557560, −6.68711405009929441386082775224, −5.38116106960755564287995484430, −4.64410265537552546462716263129, −2.53601484608829805574004729761, −1.50811148723297147777306767816, 2.21387317527355911170483025043, 3.02847063097410395674319930511, 5.11332780251907990301708107589, 5.89050131564467145905903558558, 7.07362119206095171717932064925, 7.899151886358443541808913829818, 8.773112665804411334155309790637, 10.22140462337519789790278416026, 10.80817949240338665569586186345, 11.73348302372762227057891341073

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