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Properties

Label 2-315-5.4-c3-0-37
Degree $2$
Conductor $315$
Sign $0.508 + 0.861i$
Analytic cond. $18.5856$
Root an. cond. $4.31110$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.88i·2-s − 15.9·4-s + (9.63 − 5.67i)5-s + 7i·7-s − 38.6i·8-s + (27.7 + 47.0i)10-s − 54.9·11-s + 49.7i·13-s − 34.2·14-s + 61.7·16-s − 133. i·17-s − 138.·19-s + (−153. + 90.3i)20-s − 268. i·22-s + 7.32i·23-s + ⋯
L(s)  = 1  + 1.72i·2-s − 1.98·4-s + (0.861 − 0.508i)5-s + 0.377i·7-s − 1.70i·8-s + (0.878 + 1.48i)10-s − 1.50·11-s + 1.06i·13-s − 0.653·14-s + 0.964·16-s − 1.90i·17-s − 1.67·19-s + (−1.71 + 1.01i)20-s − 2.60i·22-s + 0.0664i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.508 + 0.861i$
Analytic conductor: \(18.5856\)
Root analytic conductor: \(4.31110\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :3/2),\ 0.508 + 0.861i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.03539701976\)
\(L(\frac12)\) \(\approx\) \(0.03539701976\)
\(L(\frac{5}{2})\) 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 \( 1 + (-9.63 + 5.67i)T \)
7 \( 1 - 7iT \)
good2 \( 1 - 4.88iT - 8T^{2} \)
11 \( 1 + 54.9T + 1.33e3T^{2} \)
13 \( 1 - 49.7iT - 2.19e3T^{2} \)
17 \( 1 + 133. iT - 4.91e3T^{2} \)
19 \( 1 + 138.T + 6.85e3T^{2} \)
23 \( 1 - 7.32iT - 1.21e4T^{2} \)
29 \( 1 - 87.2T + 2.43e4T^{2} \)
31 \( 1 + 209.T + 2.97e4T^{2} \)
37 \( 1 + 67.9iT - 5.06e4T^{2} \)
41 \( 1 + 77.6T + 6.89e4T^{2} \)
43 \( 1 - 197. iT - 7.95e4T^{2} \)
47 \( 1 + 4.97iT - 1.03e5T^{2} \)
53 \( 1 + 53.0iT - 1.48e5T^{2} \)
59 \( 1 + 683.T + 2.05e5T^{2} \)
61 \( 1 + 26.8T + 2.26e5T^{2} \)
67 \( 1 - 149. iT - 3.00e5T^{2} \)
71 \( 1 + 6.15T + 3.57e5T^{2} \)
73 \( 1 + 294. iT - 3.89e5T^{2} \)
79 \( 1 - 938.T + 4.93e5T^{2} \)
83 \( 1 - 784. iT - 5.71e5T^{2} \)
89 \( 1 - 275.T + 7.04e5T^{2} \)
97 \( 1 - 1.16e3iT - 9.12e5T^{2} \)
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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

−10.89706062662984157312679036541, −9.612092038582079982672793651484, −8.999920979416752305012052040661, −8.121718404352000349504850198886, −7.08199466317272571187654760818, −6.19182733497980438631551869047, −5.22348766144781614213054108266, −4.61794223755671662328148638104, −2.35819506033864344447222079673, −0.01201604045872113484351980256, 1.73221011600200292954446563312, 2.67720172313686179506032822255, 3.77506219044873444048049878883, 5.11383293525225981341548491544, 6.25331655724225238921499817985, 7.909961974033475488823239677425, 8.852646624462222888587002065075, 10.17310435154494256458166936263, 10.55398085900160816063510746521, 10.87106070070163799453550095701

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