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Properties

Label 2-315-63.20-c1-0-14
Degree $2$
Conductor $315$
Sign $0.0155 + 0.999i$
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  + (−1.5 − 0.866i)2-s + (1.5 − 0.866i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s − 3·6-s + (2 − 1.73i)7-s + 1.73i·8-s + (1.5 − 2.59i)9-s − 1.73i·10-s + (1.5 + 0.866i)12-s + (−4.5 + 0.866i)14-s + (1.5 + 0.866i)15-s + (2.49 − 4.33i)16-s + 6·17-s + (−4.5 + 2.59i)18-s − 3.46i·19-s + ⋯
L(s)  = 1  + (−1.06 − 0.612i)2-s + (0.866 − 0.499i)3-s + (0.250 + 0.433i)4-s + (0.223 + 0.387i)5-s − 1.22·6-s + (0.755 − 0.654i)7-s + 0.612i·8-s + (0.5 − 0.866i)9-s − 0.547i·10-s + (0.433 + 0.250i)12-s + (−1.20 + 0.231i)14-s + (0.387 + 0.223i)15-s + (0.624 − 1.08i)16-s + 1.45·17-s + (−1.06 + 0.612i)18-s − 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0155 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0155 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.0155 + 0.999i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.0155 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.788029 - 0.775836i\)
\(L(\frac12)\) \(\approx\) \(0.788029 - 0.775836i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2 + 1.73i)T \)
good2 \( 1 + (1.5 + 0.866i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (7.5 - 4.33i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 + 0.866i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.46iT - 53T^{2} \)
59 \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.5 - 4.33i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 13.8iT - 73T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 15T + 89T^{2} \)
97 \( 1 + (3 + 1.73i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.34430395378260529931888874768, −10.12322967870430350168901379525, −9.785293535142005132823485287549, −8.551130740069981273702469541397, −7.88854830833867288081847641714, −7.09356048297631033372474867500, −5.50763456485427300991898805015, −3.78223144461502496500622449123, −2.38297073471721777756798976158, −1.21932808008399804972320799238, 1.77190286904786836984023836920, 3.54660982753876631616901048864, 4.90279017503110583363014756349, 6.15348354801953914161226807142, 7.70595664734042309665840030334, 8.181951593388464521575624190049, 8.861816961040088759751204567019, 9.866633713842795666636410844543, 10.33260080068755594876245351176, 11.90817083842987101794180258870

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