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Properties

Label 2-315-63.16-c1-0-1
Degree $2$
Conductor $315$
Sign $0.757 - 0.653i$
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  + (−1.17 − 2.02i)2-s + (1.32 + 1.11i)3-s + (−1.73 + 3.01i)4-s − 5-s + (0.723 − 3.98i)6-s + (−1.00 + 2.44i)7-s + 3.45·8-s + (0.492 + 2.95i)9-s + (1.17 + 2.02i)10-s − 3.25·11-s + (−5.66 + 2.03i)12-s + (0.549 + 0.951i)13-s + (6.13 − 0.830i)14-s + (−1.32 − 1.11i)15-s + (−0.565 − 0.978i)16-s + (0.763 + 1.32i)17-s + ⋯
L(s)  = 1  + (−0.827 − 1.43i)2-s + (0.762 + 0.646i)3-s + (−0.868 + 1.50i)4-s − 0.447·5-s + (0.295 − 1.62i)6-s + (−0.379 + 0.925i)7-s + 1.22·8-s + (0.164 + 0.986i)9-s + (0.369 + 0.640i)10-s − 0.981·11-s + (−1.63 + 0.586i)12-s + (0.152 + 0.263i)13-s + (1.63 − 0.222i)14-s + (−0.341 − 0.289i)15-s + (−0.141 − 0.244i)16-s + (0.185 + 0.320i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.757 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.757 - 0.653i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.757 - 0.653i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.650852 + 0.241992i\)
\(L(\frac12)\) \(\approx\) \(0.650852 + 0.241992i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.32 - 1.11i)T \)
5 \( 1 + T \)
7 \( 1 + (1.00 - 2.44i)T \)
good2 \( 1 + (1.17 + 2.02i)T + (-1 + 1.73i)T^{2} \)
11 \( 1 + 3.25T + 11T^{2} \)
13 \( 1 + (-0.549 - 0.951i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.763 - 1.32i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.06 - 5.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.30T + 23T^{2} \)
29 \( 1 + (1.48 - 2.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.04 - 3.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.94 + 8.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.222 + 0.385i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.84 + 4.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.11 + 5.39i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5.48 - 9.50i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.77 + 6.54i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.74 - 6.48i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.06 - 1.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.7T + 71T^{2} \)
73 \( 1 + (-4.96 - 8.60i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.93 + 6.80i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.19 - 14.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.440 + 0.762i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.41 + 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.51151663417653966438028727138, −10.61739745086189114981806375665, −10.06192168628233717704549906402, −8.953923984113955186407664078533, −8.585181067674718407979232592049, −7.53999004317731742714170944591, −5.55998972452206478371269089805, −4.03345481163380887088638844194, −3.07397517998066849489334595756, −2.07512794165638854412275281704, 0.59263755388891407621917586131, 2.99992020698683407018036370512, 4.66198703798663059037821745287, 6.17434910295498178153258583721, 7.07650501683074915845529202269, 7.65254876934485723425613540545, 8.390590997236806411457244267305, 9.324335932766335151827029862607, 10.20302472032007579259225494989, 11.40346870122267495087790893776

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