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Properties

Label 2-315-63.4-c1-0-16
Degree $2$
Conductor $315$
Sign $0.997 + 0.0668i$
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  + (0.845 − 1.46i)2-s + (0.216 + 1.71i)3-s + (−0.428 − 0.742i)4-s + 5-s + (2.69 + 1.13i)6-s + (−2.15 + 1.53i)7-s + 1.93·8-s + (−2.90 + 0.744i)9-s + (0.845 − 1.46i)10-s + 4.54·11-s + (1.18 − 0.897i)12-s + (1.58 − 2.74i)13-s + (0.423 + 4.45i)14-s + (0.216 + 1.71i)15-s + (2.48 − 4.31i)16-s + (−2.83 + 4.91i)17-s + ⋯
L(s)  = 1  + (0.597 − 1.03i)2-s + (0.125 + 0.992i)3-s + (−0.214 − 0.371i)4-s + 0.447·5-s + (1.10 + 0.463i)6-s + (−0.814 + 0.579i)7-s + 0.682·8-s + (−0.968 + 0.248i)9-s + (0.267 − 0.462i)10-s + 1.37·11-s + (0.341 − 0.259i)12-s + (0.439 − 0.761i)13-s + (0.113 + 1.18i)14-s + (0.0559 + 0.443i)15-s + (0.622 − 1.07i)16-s + (−0.687 + 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.997 + 0.0668i$
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: \((2,\ 315,\ (\ :1/2),\ 0.997 + 0.0668i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98042 - 0.0662575i\)
\(L(\frac12)\) \(\approx\) \(1.98042 - 0.0662575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.216 - 1.71i)T \)
5 \( 1 - T \)
7 \( 1 + (2.15 - 1.53i)T \)
good2 \( 1 + (-0.845 + 1.46i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 - 4.54T + 11T^{2} \)
13 \( 1 + (-1.58 + 2.74i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.83 - 4.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.70 - 2.94i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.0233T + 23T^{2} \)
29 \( 1 + (5.17 + 8.95i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.859 + 1.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.52 + 6.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.45 - 5.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.52 + 7.82i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.619 + 1.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.10 - 8.84i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.64 + 8.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.28 + 7.42i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.751 - 1.30i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.66T + 71T^{2} \)
73 \( 1 + (5.85 - 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.08 + 1.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.75 - 6.50i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.27 - 7.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.533 + 0.924i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.61043685663977969035871861669, −10.79526471954386955657569584324, −9.929147737413072063813239952187, −9.256041939907142544484361474163, −8.160692865178365401164548993268, −6.36761467734331828805644710497, −5.48731591875311746913376545175, −4.00885515187863325027282373724, −3.46238086423979549296866776774, −2.07845841882836516780353920313, 1.50380907369315679985085522013, 3.42066259799064281058423788173, 4.86960538297171722542178442421, 6.13912728130116970516859660437, 6.89792286358617418738870904383, 7.12369638615886067121617110107, 8.746197694310669923797102420141, 9.478546146082569958547757931389, 10.94377567308018357834632341423, 11.84963777316788711466698607278

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