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Properties

Label 2-315-63.20-c1-0-22
Degree $2$
Conductor $315$
Sign $0.986 + 0.161i$
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  + (1.02 + 0.590i)2-s + (1.63 − 0.577i)3-s + (−0.301 − 0.522i)4-s + (0.5 + 0.866i)5-s + (2.01 + 0.373i)6-s + (−2.07 − 1.63i)7-s − 3.07i·8-s + (2.33 − 1.88i)9-s + 1.18i·10-s + (5.38 + 3.10i)11-s + (−0.794 − 0.679i)12-s + (−0.534 + 0.308i)13-s + (−1.15 − 2.90i)14-s + (1.31 + 1.12i)15-s + (1.21 − 2.10i)16-s − 2.15·17-s + ⋯
L(s)  = 1  + (0.723 + 0.417i)2-s + (0.942 − 0.333i)3-s + (−0.150 − 0.261i)4-s + (0.223 + 0.387i)5-s + (0.821 + 0.152i)6-s + (−0.785 − 0.618i)7-s − 1.08i·8-s + (0.777 − 0.628i)9-s + 0.373i·10-s + (1.62 + 0.937i)11-s + (−0.229 − 0.196i)12-s + (−0.148 + 0.0855i)13-s + (−0.309 − 0.776i)14-s + (0.339 + 0.290i)15-s + (0.303 − 0.525i)16-s − 0.523·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.986 + 0.161i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.986 + 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32886 - 0.188928i\)
\(L(\frac12)\) \(\approx\) \(2.32886 - 0.188928i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.63 + 0.577i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.07 + 1.63i)T \)
good2 \( 1 + (-1.02 - 0.590i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (-5.38 - 3.10i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.534 - 0.308i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.15T + 17T^{2} \)
19 \( 1 - 4.75iT - 19T^{2} \)
23 \( 1 + (3.22 - 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.09 + 2.36i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (7.25 - 4.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.00T + 37T^{2} \)
41 \( 1 + (0.261 + 0.453i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.75 + 4.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.25 + 5.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12.7iT - 53T^{2} \)
59 \( 1 + (2.72 + 4.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-12.3 - 7.13i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.146 + 0.254i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.45iT - 71T^{2} \)
73 \( 1 - 0.406iT - 73T^{2} \)
79 \( 1 + (2.85 - 4.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.40 + 11.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.957T + 89T^{2} \)
97 \( 1 + (2.17 + 1.25i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.08074789663775505780410096365, −10.40212553536793869815252641783, −9.640372600210915714382889848029, −9.043786903301672537243441534424, −7.34225474955013917939733837341, −6.84455672145950145888578331410, −5.91540085158503266474379953591, −4.14335587192123054203205981460, −3.65371928050415633119484928151, −1.71018965016651426456957325336, 2.24213303148084406885337585131, 3.41825279240569182754456956387, 4.18103572643380417339291316116, 5.47143792403948247780523003035, 6.73260476841475996308638392331, 8.208463846215916042876069899063, 9.090056375541302969509936882794, 9.385132169539822443045645755058, 10.98418604242406778837589699487, 11.83491032314463616591622940528

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