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Properties

Label 2-315-5.3-c2-0-29
Degree $2$
Conductor $315$
Sign $-0.965 - 0.262i$
Analytic cond. $8.58312$
Root an. cond. $2.92969$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.72 − 2.72i)2-s − 10.8i·4-s + (−4.39 − 2.38i)5-s + (−1.87 + 1.87i)7-s + (−18.6 − 18.6i)8-s + (−18.4 + 5.47i)10-s − 3.42·11-s + (7.98 + 7.98i)13-s + 10.1i·14-s − 57.9·16-s + (16.5 − 16.5i)17-s − 1.38i·19-s + (−25.8 + 47.6i)20-s + (−9.31 + 9.31i)22-s + (−18.8 − 18.8i)23-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)2-s − 2.70i·4-s + (−0.879 − 0.476i)5-s + (−0.267 + 0.267i)7-s + (−2.32 − 2.32i)8-s + (−1.84 + 0.547i)10-s − 0.311·11-s + (0.613 + 0.613i)13-s + 0.727i·14-s − 3.62·16-s + (0.974 − 0.974i)17-s − 0.0726i·19-s + (−1.29 + 2.38i)20-s + (−0.423 + 0.423i)22-s + (−0.819 − 0.819i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.262i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.965 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.965 - 0.262i$
Analytic conductor: \(8.58312\)
Root analytic conductor: \(2.92969\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1),\ -0.965 - 0.262i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.317328 + 2.37947i\)
\(L(\frac12)\) \(\approx\) \(0.317328 + 2.37947i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (4.39 + 2.38i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good2 \( 1 + (-2.72 + 2.72i)T - 4iT^{2} \)
11 \( 1 + 3.42T + 121T^{2} \)
13 \( 1 + (-7.98 - 7.98i)T + 169iT^{2} \)
17 \( 1 + (-16.5 + 16.5i)T - 289iT^{2} \)
19 \( 1 + 1.38iT - 361T^{2} \)
23 \( 1 + (18.8 + 18.8i)T + 529iT^{2} \)
29 \( 1 + 45.7iT - 841T^{2} \)
31 \( 1 - 43.2T + 961T^{2} \)
37 \( 1 + (-2.18 + 2.18i)T - 1.36e3iT^{2} \)
41 \( 1 + 6.61T + 1.68e3T^{2} \)
43 \( 1 + (44.1 + 44.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (14.2 - 14.2i)T - 2.20e3iT^{2} \)
53 \( 1 + (-44.4 - 44.4i)T + 2.80e3iT^{2} \)
59 \( 1 - 17.2iT - 3.48e3T^{2} \)
61 \( 1 - 48.0T + 3.72e3T^{2} \)
67 \( 1 + (-40.9 + 40.9i)T - 4.48e3iT^{2} \)
71 \( 1 - 38.7T + 5.04e3T^{2} \)
73 \( 1 + (-66.4 - 66.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 5.30iT - 6.24e3T^{2} \)
83 \( 1 + (-62.5 - 62.5i)T + 6.88e3iT^{2} \)
89 \( 1 - 44.8iT - 7.92e3T^{2} \)
97 \( 1 + (30.5 - 30.5i)T - 9.40e3iT^{2} \)
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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

−11.41518015895013114310795656284, −10.28339970199056495491655506854, −9.483570965590929581784651678952, −8.221922870130651440146365414688, −6.61601391009470322942695476414, −5.49323444577220401045204726572, −4.50444352813708410926373201372, −3.64660647054509673418940111912, −2.45806559891100151115797120252, −0.75570084263231759557624608616, 3.21534474310435591494681870055, 3.78500660253127400846729854383, 5.05230323442528428036800185053, 6.09429587812255161132275611017, 6.93563169753740529969170739498, 7.942177026145561905079474561138, 8.357016771533141535967447331148, 10.21027496824418958743705325114, 11.41438950824909692103087454050, 12.27976243764021303748450116820

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