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Properties

Label 2-315-35.9-c1-0-2
Degree $2$
Conductor $315$
Sign $0.934 - 0.356i$
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  + (−1.34 − 0.776i)2-s + (0.204 + 0.355i)4-s + (−1.88 − 1.19i)5-s + (−0.478 + 2.60i)7-s + 2.46i·8-s + (1.61 + 3.07i)10-s + (2.21 + 3.83i)11-s − 1.73i·13-s + (2.66 − 3.12i)14-s + (2.32 − 4.02i)16-s + (2.36 − 1.36i)17-s + (−0.152 + 0.264i)19-s + (0.0376 − 0.915i)20-s − 6.87i·22-s + (6.08 + 3.51i)23-s + ⋯
L(s)  = 1  + (−0.950 − 0.548i)2-s + (0.102 + 0.177i)4-s + (−0.844 − 0.535i)5-s + (−0.180 + 0.983i)7-s + 0.872i·8-s + (0.509 + 0.972i)10-s + (0.667 + 1.15i)11-s − 0.480i·13-s + (0.711 − 0.835i)14-s + (0.581 − 1.00i)16-s + (0.573 − 0.331i)17-s + (−0.0350 + 0.0606i)19-s + (0.00841 − 0.204i)20-s − 1.46i·22-s + (1.26 + 0.732i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.934 - 0.356i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.934 - 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.573865 + 0.105645i\)
\(L(\frac12)\) \(\approx\) \(0.573865 + 0.105645i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.88 + 1.19i)T \)
7 \( 1 + (0.478 - 2.60i)T \)
good2 \( 1 + (1.34 + 0.776i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-2.21 - 3.83i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (-2.36 + 1.36i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.152 - 0.264i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.08 - 3.51i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.79T + 29T^{2} \)
31 \( 1 + (-2.64 - 4.57i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.18 - 1.83i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.71T + 41T^{2} \)
43 \( 1 - 9.71iT - 43T^{2} \)
47 \( 1 + (-1.57 - 0.908i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.48 - 0.857i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.571 - 0.989i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.77 - 8.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.26 - 4.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 + (-11.1 + 6.41i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.35 + 5.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.09iT - 83T^{2} \)
89 \( 1 + (2.03 - 3.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.87iT - 97T^{2} \)
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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

−11.68065430973539543878707101994, −10.80080753060879421706925289427, −9.458323813592195232204565507929, −9.257818521513981396243879110888, −8.136514549004675479920802036537, −7.28671601497938325057510193200, −5.64458848648665916608838013206, −4.64417992514253459008819984240, −2.98983776816962518657238932961, −1.36960465278444041911343575474, 0.66649797788646870767222291466, 3.40423287425680787396726365236, 4.17211301925413647405630253334, 6.18091762005368408471840137084, 7.05734507838956761402769648862, 7.74202829023759922154098671568, 8.670145978299532175866671887061, 9.539669606972393088823823723778, 10.68977326710736130904180424983, 11.25603905095156715573640218643

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