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Properties

Label 2-315-35.23-c2-0-9
Degree $2$
Conductor $315$
Sign $0.177 + 0.984i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.955 − 3.56i)2-s + (−8.33 + 4.81i)4-s + (3.70 − 3.36i)5-s + (3.07 + 6.28i)7-s + (14.6 + 14.6i)8-s + (−15.5 − 9.98i)10-s + (4.09 + 7.09i)11-s + (14.0 + 14.0i)13-s + (19.4 − 16.9i)14-s + (19.0 − 32.9i)16-s + (6.76 + 1.81i)17-s + (18.2 + 10.5i)19-s + (−14.6 + 45.8i)20-s + (21.3 − 21.3i)22-s + (−31.4 + 8.43i)23-s + ⋯
L(s)  = 1  + (−0.477 − 1.78i)2-s + (−2.08 + 1.20i)4-s + (0.740 − 0.672i)5-s + (0.439 + 0.898i)7-s + (1.83 + 1.83i)8-s + (−1.55 − 0.998i)10-s + (0.372 + 0.644i)11-s + (1.08 + 1.08i)13-s + (1.39 − 1.21i)14-s + (1.18 − 2.06i)16-s + (0.397 + 0.106i)17-s + (0.959 + 0.554i)19-s + (−0.733 + 2.29i)20-s + (0.971 − 0.971i)22-s + (−1.36 + 0.366i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.177 + 0.984i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ 0.177 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.06830 - 0.893272i\)
\(L(\frac12)\) \(\approx\) \(1.06830 - 0.893272i\)
\(L(2)\) 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 \( 1 + (-3.70 + 3.36i)T \)
7 \( 1 + (-3.07 - 6.28i)T \)
good2 \( 1 + (0.955 + 3.56i)T + (-3.46 + 2i)T^{2} \)
11 \( 1 + (-4.09 - 7.09i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-14.0 - 14.0i)T + 169iT^{2} \)
17 \( 1 + (-6.76 - 1.81i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (-18.2 - 10.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (31.4 - 8.43i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 22.1iT - 841T^{2} \)
31 \( 1 + (-13.7 - 23.7i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (4.01 + 14.9i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 0.496T + 1.68e3T^{2} \)
43 \( 1 + (33.4 + 33.4i)T + 1.84e3iT^{2} \)
47 \( 1 + (-6.46 - 24.1i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (9.20 - 34.3i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (15.5 - 9.00i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-13.4 + 23.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-5.37 - 1.44i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 105.T + 5.04e3T^{2} \)
73 \( 1 + (-21.8 + 81.6i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (-86.5 - 49.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (95.6 + 95.6i)T + 6.88e3iT^{2} \)
89 \( 1 + (-54.3 - 31.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-59.0 + 59.0i)T - 9.40e3iT^{2} \)
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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

−11.41872663713223392626070493125, −10.20337709301819582233568421803, −9.506097156761866591837341538174, −8.862487183228178172460447486161, −8.021142191203681808768713321147, −6.03697652899669115146514397026, −4.80825871601604978080779277064, −3.70629962757465759081306734567, −2.10662282142368116659141489381, −1.40314271111743114312771618798, 0.953253100080584595129020724984, 3.58996691261750752089445209053, 5.13503644169852664313215405850, 6.01764872168575730633869038470, 6.76675416434358996574656800313, 7.78181084357349277089572743958, 8.445717901396821843567276967820, 9.655760187317755404925336702943, 10.32754987558978583355019406132, 11.31228986586973887986186129158

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