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Properties

Label 2-315-315.178-c1-0-17
Degree $2$
Conductor $315$
Sign $-0.310 - 0.950i$
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  + (−1.36 + 1.36i)2-s + (0.866 + 1.5i)3-s − 1.73i·4-s + (1.86 − 1.23i)5-s + (−3.23 − 0.866i)6-s + (1.73 − 2i)7-s + (−0.366 − 0.366i)8-s + (−1.5 + 2.59i)9-s + (−0.866 + 4.23i)10-s + (2.73 + 4.73i)11-s + (2.59 − 1.49i)12-s + (3.73 + i)13-s + (0.366 + 5.09i)14-s + (3.46 + 1.73i)15-s + 4.46·16-s + (−2.73 + 0.732i)17-s + ⋯
L(s)  = 1  + (−0.965 + 0.965i)2-s + (0.499 + 0.866i)3-s − 0.866i·4-s + (0.834 − 0.550i)5-s + (−1.31 − 0.353i)6-s + (0.654 − 0.755i)7-s + (−0.129 − 0.129i)8-s + (−0.5 + 0.866i)9-s + (−0.273 + 1.33i)10-s + (0.823 + 1.42i)11-s + (0.749 − 0.433i)12-s + (1.03 + 0.277i)13-s + (0.0978 + 1.36i)14-s + (0.894 + 0.447i)15-s + 1.11·16-s + (−0.662 + 0.177i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.310 - 0.950i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670025 + 0.923673i\)
\(L(\frac12)\) \(\approx\) \(0.670025 + 0.923673i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (-1.86 + 1.23i)T \)
7 \( 1 + (-1.73 + 2i)T \)
good2 \( 1 + (1.36 - 1.36i)T - 2iT^{2} \)
11 \( 1 + (-2.73 - 4.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.73 - i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.73 - 0.732i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.63 + 2.83i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.59 - 1.23i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-6 - 3.46i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.196iT - 31T^{2} \)
37 \( 1 + (4.09 + 1.09i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.19 - 1.26i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.5 + 0.401i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (9.29 + 9.29i)T + 47iT^{2} \)
53 \( 1 + (-6.09 + 1.63i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 - 11.3iT - 61T^{2} \)
67 \( 1 + (-0.901 + 0.901i)T - 67iT^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (2.36 + 8.83i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 - 4.53iT - 79T^{2} \)
83 \( 1 + (0.366 + 1.36i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (4.59 + 7.96i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.83 + 2.09i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.73792256023878612163197879753, −10.38565749884196074414456826367, −9.917758045092013001335113515174, −8.831824589764631438650696009944, −8.554386423439853923207165156219, −7.25156438808280564175187083999, −6.33909613597215729564994022609, −4.92460158535911680887916839206, −3.98179718612414852703189367742, −1.73387280747798962661615550270, 1.29998517848871088557031381670, 2.32739623429911426232569194214, 3.39888968716194751576106329738, 5.86984694786492972317024534038, 6.37795236340172816425643001488, 8.181406966916716077368174745918, 8.576539158940848120690385286907, 9.383061826679009794990064163425, 10.52692252064994418284508736035, 11.36045243243703341621927864920

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