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Properties

Label 2-315-15.2-c1-0-9
Degree $2$
Conductor $315$
Sign $-0.235 + 0.971i$
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  + (1.62 − 1.62i)2-s − 3.25i·4-s + (1.83 − 1.28i)5-s + (−0.707 − 0.707i)7-s + (−2.02 − 2.02i)8-s + (0.894 − 5.04i)10-s + 3.03i·11-s + (−2.54 + 2.54i)13-s − 2.29·14-s − 0.0700·16-s + (−2.70 + 2.70i)17-s − 6.63i·19-s + (−4.16 − 5.96i)20-s + (4.91 + 4.91i)22-s + (2.99 + 2.99i)23-s + ⋯
L(s)  = 1  + (1.14 − 1.14i)2-s − 1.62i·4-s + (0.819 − 0.572i)5-s + (−0.267 − 0.267i)7-s + (−0.717 − 0.717i)8-s + (0.282 − 1.59i)10-s + 0.914i·11-s + (−0.704 + 0.704i)13-s − 0.612·14-s − 0.0175·16-s + (−0.655 + 0.655i)17-s − 1.52i·19-s + (−0.931 − 1.33i)20-s + (1.04 + 1.04i)22-s + (0.623 + 0.623i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.235 + 0.971i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52296 - 1.93546i\)
\(L(\frac12)\) \(\approx\) \(1.52296 - 1.93546i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.83 + 1.28i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (-1.62 + 1.62i)T - 2iT^{2} \)
11 \( 1 - 3.03iT - 11T^{2} \)
13 \( 1 + (2.54 - 2.54i)T - 13iT^{2} \)
17 \( 1 + (2.70 - 2.70i)T - 17iT^{2} \)
19 \( 1 + 6.63iT - 19T^{2} \)
23 \( 1 + (-2.99 - 2.99i)T + 23iT^{2} \)
29 \( 1 + 5.10T + 29T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 + (-6.68 - 6.68i)T + 37iT^{2} \)
41 \( 1 - 7.36iT - 41T^{2} \)
43 \( 1 + (-1.74 + 1.74i)T - 43iT^{2} \)
47 \( 1 + (0.173 - 0.173i)T - 47iT^{2} \)
53 \( 1 + (-8.45 - 8.45i)T + 53iT^{2} \)
59 \( 1 - 4.60T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + (8.00 + 8.00i)T + 67iT^{2} \)
71 \( 1 + 5.47iT - 71T^{2} \)
73 \( 1 + (5.58 - 5.58i)T - 73iT^{2} \)
79 \( 1 + 16.1iT - 79T^{2} \)
83 \( 1 + (0.998 + 0.998i)T + 83iT^{2} \)
89 \( 1 - 3.00T + 89T^{2} \)
97 \( 1 + (8.52 + 8.52i)T + 97iT^{2} \)
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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

−11.55905118821308880136208279479, −10.68191924488616879913797649850, −9.705010175938218326604195223520, −9.085580550562797373710541386462, −7.30137356236749765803313465832, −6.12317033748109682102667901059, −4.89821476498311333987173079234, −4.34483052283103508148626528196, −2.73808679088739839102645917657, −1.66074386388108311923597422288, 2.66363127018185519573059355125, 3.84595622880058365022689380441, 5.39043089101678447887875828975, 5.82781441443856575994035784825, 6.84951489298210346429740677360, 7.69922515065180893902169114540, 8.935718222755120978775604696008, 10.07410434511743140811741416866, 11.07853265299570371973323138087, 12.37503804937750041426962402251

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