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Properties

Label 2-315-63.41-c1-0-11
Degree $2$
Conductor $315$
Sign $0.992 + 0.125i$
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 + (2.45 − 0.981i)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 + (1.93 − 2.45i)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 + (0.928 − 0.370i)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 + (0.517 − 0.656i)14-s + (0.339 − 0.290i)15-s + (0.303 + 0.525i)16-s + 0.523·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.992 + 0.125i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.992 + 0.125i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49813 - 0.0940342i\)
\(L(\frac12)\) \(\approx\) \(1.49813 - 0.0940342i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.45 + 0.981i)T \)
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

−11.75215363033430486901706653742, −11.16503494559469059242281599231, −10.20743775663091669243073458300, −8.600142074698733050186984034345, −7.78379549559176327589250352960, −6.56639439907243686396529609769, −5.57953641378185039909109749135, −4.39702487684075668351190625826, −3.55717134309919937094126418092, −1.54721958669742331942900128227, 1.29081719078327281684501173156, 4.08732541013467403382416327335, 4.60335214315724303088339366395, 5.61573781909639049805616014745, 6.47606558734841673022241712577, 7.54286536123001493250329130839, 9.130918373263730431453377115391, 9.724829519731045915376985274041, 10.96264629675955563881528695799, 11.92104688714376212765718876519

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