Properties

Label 2-315-63.4-c1-0-1
Degree $2$
Conductor $315$
Sign $-0.975 - 0.220i$
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  + (−0.651 + 1.12i)2-s − 1.73i·3-s + (0.151 + 0.262i)4-s − 5-s + (1.95 + 1.12i)6-s + (−2 + 1.73i)7-s − 3·8-s − 2.99·9-s + (0.651 − 1.12i)10-s + (0.454 − 0.262i)12-s + (−0.802 + 1.39i)13-s + (−0.651 − 3.38i)14-s + 1.73i·15-s + (1.65 − 2.86i)16-s + (−2.80 + 4.85i)17-s + (1.95 − 3.38i)18-s + ⋯
L(s)  = 1  + (−0.460 + 0.797i)2-s − 0.999i·3-s + (0.0756 + 0.131i)4-s − 0.447·5-s + (0.797 + 0.460i)6-s + (−0.755 + 0.654i)7-s − 1.06·8-s − 0.999·9-s + (0.205 − 0.356i)10-s + (0.131 − 0.0756i)12-s + (−0.222 + 0.385i)13-s + (−0.174 − 0.904i)14-s + 0.447i·15-s + (0.412 − 0.715i)16-s + (−0.679 + 1.17i)17-s + (0.460 − 0.797i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.975 - 0.220i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0417545 + 0.374084i\)
\(L(\frac12)\) \(\approx\) \(0.0417545 + 0.374084i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
5 \( 1 + T \)
7 \( 1 + (2 - 1.73i)T \)
good2 \( 1 + (0.651 - 1.12i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (0.802 - 1.39i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.80 - 4.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.80 - 3.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.21T + 23T^{2} \)
29 \( 1 + (4.10 + 7.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.80 - 3.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.80 - 3.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.10 + 3.64i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.10 + 7.11i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.197 - 0.341i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.80 - 10.0i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.10 - 3.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.10 + 3.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-1.80 + 3.12i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.40 - 11.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.10 - 12.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.80 + 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.09155195942118212728288649218, −11.56345378474093334163637602186, −9.993758634031796420529661107952, −8.812978337389485324652056456340, −8.214514261450284633286406331420, −7.32193701482572819909259005595, −6.39903077031519178506375107982, −5.78664160066195044118359380138, −3.68808706229248503943486448542, −2.29974501951793555304703836069, 0.29050338361755993190209605357, 2.70003123804498082673022067156, 3.68144977542744759398914833723, 4.97021423051491892564764777273, 6.24669785142458636759314908891, 7.47845908846250594201841338284, 8.921941721025809459303442541822, 9.554494574192334160190141955959, 10.29936406103094609581506346944, 11.09595406764655936960089152424

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