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Properties

Label 2-315-105.104-c1-0-2
Degree $2$
Conductor $315$
Sign $-0.0605 - 0.998i$
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  − 0.741·2-s − 1.44·4-s + (2.05 + 0.880i)5-s + (−1.24 + 2.33i)7-s + 2.55·8-s + (−1.52 − 0.653i)10-s − 1.41i·11-s − 5.54·13-s + (0.923 − 1.73i)14-s + 1.00·16-s + 6.07i·17-s + 7.12i·19-s + (−2.97 − 1.27i)20-s + 1.04i·22-s + 4.78·23-s + ⋯
L(s)  = 1  − 0.524·2-s − 0.724·4-s + (0.919 + 0.393i)5-s + (−0.470 + 0.882i)7-s + 0.904·8-s + (−0.482 − 0.206i)10-s − 0.426i·11-s − 1.53·13-s + (0.246 − 0.462i)14-s + 0.250·16-s + 1.47i·17-s + 1.63i·19-s + (−0.666 − 0.285i)20-s + 0.223i·22-s + 0.997·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0605 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0605 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0605 - 0.998i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (314, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.0605 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.520764 + 0.553306i\)
\(L(\frac12)\) \(\approx\) \(0.520764 + 0.553306i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-2.05 - 0.880i)T \)
7 \( 1 + (1.24 - 2.33i)T \)
good2 \( 1 + 0.741T + 2T^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 5.54T + 13T^{2} \)
17 \( 1 - 6.07iT - 17T^{2} \)
19 \( 1 - 7.12iT - 19T^{2} \)
23 \( 1 - 4.78T + 23T^{2} \)
29 \( 1 - 5.51iT - 29T^{2} \)
31 \( 1 + 1.30iT - 31T^{2} \)
37 \( 1 - 2.57iT - 37T^{2} \)
41 \( 1 + 5.95T + 41T^{2} \)
43 \( 1 + 6.76iT - 43T^{2} \)
47 \( 1 + 7.83iT - 47T^{2} \)
53 \( 1 - 9.90T + 53T^{2} \)
59 \( 1 - 1.84T + 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 + 7.23iT - 67T^{2} \)
71 \( 1 + 8.34iT - 71T^{2} \)
73 \( 1 - 0.559T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 7.83iT - 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 5.54T + 97T^{2} \)
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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

−12.05491157164786813251723843818, −10.42059704761806519465176967086, −10.11308999789974941121617779522, −9.096869353376269215128344243601, −8.422630884666580969372456012876, −7.14695334233535637546721009253, −5.90894146696835284024087085699, −5.11945409534802490449963989606, −3.44842211959916936799481286680, −1.91237660527723420637170265497, 0.67174257575544012299397704672, 2.64664613637685769588946845608, 4.58426730363756819115560733408, 5.09915696969032298299949717588, 6.84710461995380340840404517729, 7.50720801820498909261616726774, 8.936966040868849391676543925831, 9.641612369534738464774628952722, 10.00909923235861433774252093157, 11.22720599103404737740470813939

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