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Properties

Label 2-315-63.4-c1-0-14
Degree $2$
Conductor $315$
Sign $0.989 + 0.147i$
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  + (−0.926 + 1.60i)2-s + (−0.957 − 1.44i)3-s + (−0.715 − 1.24i)4-s + 5-s + (3.20 − 0.199i)6-s + (−2.45 − 0.989i)7-s − 1.05·8-s + (−1.16 + 2.76i)9-s + (−0.926 + 1.60i)10-s + 5.26·11-s + (−1.10 + 2.22i)12-s + (1.03 − 1.80i)13-s + (3.86 − 3.02i)14-s + (−0.957 − 1.44i)15-s + (2.40 − 4.16i)16-s + (3.07 − 5.33i)17-s + ⋯
L(s)  = 1  + (−0.654 + 1.13i)2-s + (−0.552 − 0.833i)3-s + (−0.357 − 0.620i)4-s + 0.447·5-s + (1.30 − 0.0813i)6-s + (−0.927 − 0.373i)7-s − 0.372·8-s + (−0.388 + 0.921i)9-s + (−0.292 + 0.507i)10-s + 1.58·11-s + (−0.318 + 0.641i)12-s + (0.288 − 0.499i)13-s + (1.03 − 0.807i)14-s + (−0.247 − 0.372i)15-s + (0.601 − 1.04i)16-s + (0.746 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.147i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.989 + 0.147i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.989 + 0.147i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.741200 - 0.0548597i\)
\(L(\frac12)\) \(\approx\) \(0.741200 - 0.0548597i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.957 + 1.44i)T \)
5 \( 1 - T \)
7 \( 1 + (2.45 + 0.989i)T \)
good2 \( 1 + (0.926 - 1.60i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 - 5.26T + 11T^{2} \)
13 \( 1 + (-1.03 + 1.80i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.07 + 5.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.23 + 5.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 4.54T + 23T^{2} \)
29 \( 1 + (-1.69 - 2.93i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.767 - 1.32i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.91 - 3.31i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.41 + 4.17i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.77 + 9.99i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.0703 - 0.121i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.80 - 6.58i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.61 + 9.71i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.95 - 3.37i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.50 + 7.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.90T + 71T^{2} \)
73 \( 1 + (3.71 - 6.44i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.05 + 8.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.05 - 13.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.0948 - 0.164i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.39 - 4.14i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.75432026843127211991312672288, −10.59547378716930872697281965010, −9.373877348008967512552720039880, −8.810734057123234512481075231951, −7.44947076398348415540924393434, −6.75694216500900534022256954316, −6.28929850855325646003434054752, −5.10016284538871255848751090949, −3.04992499046062170337903142702, −0.798096561829946419571711243841, 1.45708617055183568846105679576, 3.24036011931101306131193991473, 4.12450366290243677212699168033, 6.10380075962208510190531893980, 6.25936836112773489372301718307, 8.539854009413781096354436688323, 9.313937077881890758482151272817, 9.876103744278850064463703126737, 10.60577812846149314855029476401, 11.56409084062596024238225634344

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