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Properties

Label 2-315-63.47-c1-0-18
Degree $2$
Conductor $315$
Sign $0.997 + 0.0682i$
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  + (−1.16 + 0.673i)2-s + (1.48 − 0.887i)3-s + (−0.0930 + 0.161i)4-s + 5-s + (−1.13 + 2.03i)6-s + (2.59 − 0.537i)7-s − 2.94i·8-s + (1.42 − 2.63i)9-s + (−1.16 + 0.673i)10-s − 4.70i·11-s + (0.00457 + 0.322i)12-s + (−4.03 + 2.33i)13-s + (−2.65 + 2.37i)14-s + (1.48 − 0.887i)15-s + (1.79 + 3.11i)16-s + (2.39 + 4.14i)17-s + ⋯
L(s)  = 1  + (−0.824 + 0.476i)2-s + (0.858 − 0.512i)3-s + (−0.0465 + 0.0805i)4-s + 0.447·5-s + (−0.464 + 0.831i)6-s + (0.979 − 0.203i)7-s − 1.04i·8-s + (0.475 − 0.879i)9-s + (−0.368 + 0.212i)10-s − 1.41i·11-s + (0.00132 + 0.0930i)12-s + (−1.12 + 0.646i)13-s + (−0.710 + 0.633i)14-s + (0.384 − 0.229i)15-s + (0.449 + 0.777i)16-s + (0.579 + 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.997 + 0.0682i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.997 + 0.0682i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23975 - 0.0423776i\)
\(L(\frac12)\) \(\approx\) \(1.23975 - 0.0423776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.48 + 0.887i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.59 + 0.537i)T \)
good2 \( 1 + (1.16 - 0.673i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + 4.70iT - 11T^{2} \)
13 \( 1 + (4.03 - 2.33i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.39 - 4.14i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.17 - 1.25i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.40iT - 23T^{2} \)
29 \( 1 + (0.970 + 0.560i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.64 - 3.83i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.507 - 0.878i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.36 - 7.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.74 - 4.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.13 + 10.6i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.6 - 6.74i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.57 - 4.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.20 + 1.27i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.70 - 9.88i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.31iT - 71T^{2} \)
73 \( 1 + (3.32 - 1.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.20 + 5.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.81 - 3.14i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.794 + 1.37i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.58 + 2.07i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.72095633178830147643382067420, −10.35581839350508716451162176250, −9.537864715278189061231611966454, −8.420070923722357802364608488518, −8.198633034181025943940561259171, −7.12532294964035787040077647335, −6.13078628241094958721617957412, −4.43332063043604411798849502426, −3.02707239210427183637945514835, −1.31443355212335895893758011078, 1.74499673350813068409574386368, 2.73743902956596949709555872501, 4.75170329677770329409169791037, 5.25738443338628572817476825004, 7.47222703233805518317459083388, 7.939617131596784123027726823191, 9.321077009657761639213083742237, 9.614319601710008804729525231214, 10.37299915535266336171109018655, 11.42112624693027390589289834903

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