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Properties

Label 2-315-35.4-c1-0-7
Degree $2$
Conductor $315$
Sign $0.758 - 0.652i$
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  + (−0.866 + 0.5i)2-s + (−0.500 + 0.866i)4-s + (2.23 − 0.133i)5-s + (2.59 − 0.5i)7-s − 3i·8-s + (−1.86 + 1.23i)10-s − 2i·13-s + (−2 + 1.73i)14-s + (0.500 + 0.866i)16-s + (1.73 + i)17-s + (3 + 5.19i)19-s + (−1.00 + 1.99i)20-s + (−2.59 + 1.5i)23-s + (4.96 − 0.598i)25-s + (1 + 1.73i)26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.250 + 0.433i)4-s + (0.998 − 0.0599i)5-s + (0.981 − 0.188i)7-s − 1.06i·8-s + (−0.590 + 0.389i)10-s − 0.554i·13-s + (−0.534 + 0.462i)14-s + (0.125 + 0.216i)16-s + (0.420 + 0.242i)17-s + (0.688 + 1.19i)19-s + (−0.223 + 0.447i)20-s + (−0.541 + 0.312i)23-s + (0.992 − 0.119i)25-s + (0.196 + 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.652i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.758 - 0.652i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.758 - 0.652i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07934 + 0.400302i\)
\(L(\frac12)\) \(\approx\) \(1.07934 + 0.400302i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.133i)T \)
7 \( 1 + (-2.59 + 0.5i)T \)
good2 \( 1 + (0.866 - 0.5i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + (-1.73 - i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.92 - 4i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.19 + 3i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (5.19 + 3i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1 + 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 16iT - 97T^{2} \)
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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

−11.88560330338452182633165092058, −10.38917643622884758791187504131, −9.976898840984173998864587690302, −8.752599937515953523157801205463, −8.109634480815861904650392522973, −7.16740253502427371740506492359, −5.91030439690509573297726387943, −4.83917553324183279065437108278, −3.36792355653108365370266870112, −1.48958335002481655634818506321, 1.35675966927314170876715232013, 2.53247835130626565846260934077, 4.72330931477009304620142931760, 5.44474803353433179002083624620, 6.65379077633378833328611523862, 8.040019877867038931166147554489, 8.987107171476530975261749631874, 9.635883263673471045990917869201, 10.55996158703446973952465231268, 11.29439533609571104838711051467

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