Properties

Label 2-315-15.8-c1-0-0
Degree $2$
Conductor $315$
Sign $0.837 - 0.545i$
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  + (−1.62 − 1.62i)2-s + 3.25i·4-s + (−1.83 − 1.28i)5-s + (−0.707 + 0.707i)7-s + (2.02 − 2.02i)8-s + (0.894 + 5.04i)10-s + 3.03i·11-s + (−2.54 − 2.54i)13-s + 2.29·14-s − 0.0700·16-s + (2.70 + 2.70i)17-s + 6.63i·19-s + (4.16 − 5.96i)20-s + (4.91 − 4.91i)22-s + (−2.99 + 2.99i)23-s + ⋯
L(s)  = 1  + (−1.14 − 1.14i)2-s + 1.62i·4-s + (−0.819 − 0.572i)5-s + (−0.267 + 0.267i)7-s + (0.717 − 0.717i)8-s + (0.282 + 1.59i)10-s + 0.914i·11-s + (−0.704 − 0.704i)13-s + 0.612·14-s − 0.0175·16-s + (0.655 + 0.655i)17-s + 1.52i·19-s + (0.931 − 1.33i)20-s + (1.04 − 1.04i)22-s + (−0.623 + 0.623i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.837 - 0.545i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.837 - 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.330727 + 0.0982003i\)
\(L(\frac12)\) \(\approx\) \(0.330727 + 0.0982003i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 + 1.28i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + (1.62 + 1.62i)T + 2iT^{2} \)
11 \( 1 - 3.03iT - 11T^{2} \)
13 \( 1 + (2.54 + 2.54i)T + 13iT^{2} \)
17 \( 1 + (-2.70 - 2.70i)T + 17iT^{2} \)
19 \( 1 - 6.63iT - 19T^{2} \)
23 \( 1 + (2.99 - 2.99i)T - 23iT^{2} \)
29 \( 1 - 5.10T + 29T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 + (-6.68 + 6.68i)T - 37iT^{2} \)
41 \( 1 - 7.36iT - 41T^{2} \)
43 \( 1 + (-1.74 - 1.74i)T + 43iT^{2} \)
47 \( 1 + (-0.173 - 0.173i)T + 47iT^{2} \)
53 \( 1 + (8.45 - 8.45i)T - 53iT^{2} \)
59 \( 1 + 4.60T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + (8.00 - 8.00i)T - 67iT^{2} \)
71 \( 1 + 5.47iT - 71T^{2} \)
73 \( 1 + (5.58 + 5.58i)T + 73iT^{2} \)
79 \( 1 - 16.1iT - 79T^{2} \)
83 \( 1 + (-0.998 + 0.998i)T - 83iT^{2} \)
89 \( 1 + 3.00T + 89T^{2} \)
97 \( 1 + (8.52 - 8.52i)T - 97iT^{2} \)
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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

−11.78226493979865345451984868776, −10.65153010923698800051461041239, −9.895338391627973218411965085766, −9.197172327874433752247338070176, −7.937869031906812190628700345459, −7.71534199280287124635451334139, −5.77030355930412003375270228756, −4.23716453270412341984524680776, −3.04210148246363666750436109781, −1.49801171794525718295267455815, 0.37742491945095649180632135366, 3.04516889736231943912602408026, 4.67820246973378988042803529494, 6.18248735422642151666625448303, 6.97100513520335568492715620795, 7.66791827838698204480082438214, 8.598941656644384274421388455870, 9.465506913704390596730208301599, 10.41200277858363145776239443312, 11.32583188427847228644650248477

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