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Properties

Label 2-315-315.2-c1-0-22
Degree $2$
Conductor $315$
Sign $0.902 + 0.430i$
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  + (−0.534 − 0.143i)2-s + (1.61 + 0.623i)3-s + (−1.46 − 0.846i)4-s + (2.22 + 0.203i)5-s + (−0.774 − 0.564i)6-s + (−1.18 − 2.36i)7-s + (1.44 + 1.44i)8-s + (2.22 + 2.01i)9-s + (−1.16 − 0.427i)10-s − 4.16i·11-s + (−1.84 − 2.28i)12-s + (3.13 + 0.839i)13-s + (0.294 + 1.43i)14-s + (3.47 + 1.71i)15-s + (1.12 + 1.95i)16-s + (0.204 + 0.0547i)17-s + ⋯
L(s)  = 1  + (−0.377 − 0.101i)2-s + (0.932 + 0.359i)3-s + (−0.733 − 0.423i)4-s + (0.995 + 0.0909i)5-s + (−0.316 − 0.230i)6-s + (−0.447 − 0.894i)7-s + (0.510 + 0.510i)8-s + (0.740 + 0.671i)9-s + (−0.367 − 0.135i)10-s − 1.25i·11-s + (−0.531 − 0.659i)12-s + (0.868 + 0.232i)13-s + (0.0786 + 0.383i)14-s + (0.896 + 0.443i)15-s + (0.282 + 0.488i)16-s + (0.0495 + 0.0132i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.902 + 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41015 - 0.318731i\)
\(L(\frac12)\) \(\approx\) \(1.41015 - 0.318731i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.61 - 0.623i)T \)
5 \( 1 + (-2.22 - 0.203i)T \)
7 \( 1 + (1.18 + 2.36i)T \)
good2 \( 1 + (0.534 + 0.143i)T + (1.73 + i)T^{2} \)
11 \( 1 + 4.16iT - 11T^{2} \)
13 \( 1 + (-3.13 - 0.839i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-0.204 - 0.0547i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.17 + 0.678i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.134 - 0.134i)T + 23iT^{2} \)
29 \( 1 + (-2.87 + 4.98i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.19 - 8.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.73 + 1.26i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (3.39 - 1.95i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.609 - 2.27i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (9.29 + 2.49i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.78 + 6.64i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.744 - 1.28i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.95 - 5.12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.2 - 3.00i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 8.70iT - 71T^{2} \)
73 \( 1 + (10.7 + 2.88i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (11.8 - 6.84i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.88 + 1.30i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (8.20 - 14.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.62 + 1.23i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.07356499672369913741112078260, −10.39363269710311994927558717401, −9.717457815777126930621662427756, −8.855490037113972479395553892806, −8.236917284396101549322289649775, −6.75484037473746445000445546978, −5.60781642707655951256081148034, −4.29833686751165421391876625302, −3.15069176299368407467912692705, −1.36487317256820440267479414715, 1.78969571982192908650125315512, 3.13504987992576959606930218995, 4.52973601853292139753963356870, 5.94810249440871422928532778378, 7.07564225533247811292107461466, 8.162354339733847923250712886826, 9.027767139590404306598676293042, 9.506310514108640252691178098924, 10.31592838664744804927327980140, 12.09669138389831622835701304633

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