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Properties

Label 2-315-105.89-c1-0-13
Degree $2$
Conductor $315$
Sign $0.0424 + 0.999i$
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  + (0.659 − 1.14i)2-s + (0.130 + 0.226i)4-s + (0.729 − 2.11i)5-s + (−2.12 − 1.57i)7-s + 2.98·8-s + (−1.93 − 2.22i)10-s + (2.08 − 1.20i)11-s − 1.69·13-s + (−3.19 + 1.38i)14-s + (1.70 − 2.95i)16-s + (−0.831 + 0.480i)17-s + (3.56 + 2.05i)19-s + (0.574 − 0.111i)20-s − 3.17i·22-s + (2.88 − 4.99i)23-s + ⋯
L(s)  = 1  + (0.466 − 0.807i)2-s + (0.0654 + 0.113i)4-s + (0.326 − 0.945i)5-s + (−0.803 − 0.595i)7-s + 1.05·8-s + (−0.610 − 0.704i)10-s + (0.629 − 0.363i)11-s − 0.469·13-s + (−0.855 + 0.371i)14-s + (0.425 − 0.737i)16-s + (−0.201 + 0.116i)17-s + (0.818 + 0.472i)19-s + (0.128 − 0.0248i)20-s − 0.677i·22-s + (0.601 − 1.04i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0424 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0424 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0424 + 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.0424 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29366 - 1.23986i\)
\(L(\frac12)\) \(\approx\) \(1.29366 - 1.23986i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.729 + 2.11i)T \)
7 \( 1 + (2.12 + 1.57i)T \)
good2 \( 1 + (-0.659 + 1.14i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-2.08 + 1.20i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.69T + 13T^{2} \)
17 \( 1 + (0.831 - 0.480i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.56 - 2.05i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.88 + 4.99i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.56iT - 29T^{2} \)
31 \( 1 + (7.58 - 4.37i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.44 - 1.98i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.87T + 41T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 + (8.75 + 5.05i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.35 - 11.0i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.38 + 5.86i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.98 - 1.14i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.15 - 2.39i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.24iT - 71T^{2} \)
73 \( 1 + (-7.34 - 12.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.892 + 1.54i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.62iT - 83T^{2} \)
89 \( 1 + (0.220 - 0.382i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−11.58221793366150700272227835498, −10.61980104670706628671401566003, −9.717404329568350013901994126622, −8.802859592554343244136516495623, −7.57471702544286014823947858757, −6.51730974734709937070543502234, −5.11301133473675419857213198601, −4.06892980091378762799563735818, −3.00441774760159485551886699822, −1.32403804819933185887476958276, 2.21931899625330541728950572378, 3.67135556284920355103555665447, 5.23152524864495444850102352461, 6.06841457386007965063765567447, 6.92387219755237639683233700781, 7.56525160640978410156360116999, 9.321654131808812127390265819419, 9.836175044335833946662356551166, 11.01575356093756174358778654527, 11.82696542044774384052701244347

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