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Properties

Label 2-315-63.59-c1-0-31
Degree $2$
Conductor $315$
Sign $-0.694 + 0.719i$
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  + (0.403 + 0.232i)2-s + (0.625 − 1.61i)3-s + (−0.891 − 1.54i)4-s − 5-s + (0.628 − 0.505i)6-s + (−2.59 + 0.507i)7-s − 1.76i·8-s + (−2.21 − 2.01i)9-s + (−0.403 − 0.232i)10-s − 0.201i·11-s + (−3.05 + 0.474i)12-s + (0.459 + 0.265i)13-s + (−1.16 − 0.400i)14-s + (−0.625 + 1.61i)15-s + (−1.37 + 2.37i)16-s + (2.13 − 3.70i)17-s + ⋯
L(s)  = 1  + (0.285 + 0.164i)2-s + (0.360 − 0.932i)3-s + (−0.445 − 0.772i)4-s − 0.447·5-s + (0.256 − 0.206i)6-s + (−0.981 + 0.191i)7-s − 0.623i·8-s + (−0.739 − 0.673i)9-s + (−0.127 − 0.0736i)10-s − 0.0606i·11-s + (−0.880 + 0.137i)12-s + (0.127 + 0.0735i)13-s + (−0.311 − 0.106i)14-s + (−0.161 + 0.417i)15-s + (−0.343 + 0.594i)16-s + (0.518 − 0.898i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.694 + 0.719i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.694 + 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408248 - 0.961682i\)
\(L(\frac12)\) \(\approx\) \(0.408248 - 0.961682i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.625 + 1.61i)T \)
5 \( 1 + T \)
7 \( 1 + (2.59 - 0.507i)T \)
good2 \( 1 + (-0.403 - 0.232i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + 0.201iT - 11T^{2} \)
13 \( 1 + (-0.459 - 0.265i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.13 + 3.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0968 - 0.0559i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.40iT - 23T^{2} \)
29 \( 1 + (-1.21 + 0.702i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.59 + 2.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.817 + 1.41i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.47 + 7.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.56 - 7.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.76 + 8.24i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.28 + 3.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.66 - 2.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.46 - 5.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.95 + 6.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 + (3.84 + 2.21i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.62 - 14.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.64 + 4.57i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.74 + 4.75i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.347 + 0.200i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.54007238424266590446266923581, −10.25192297910929003821529365788, −9.342576622357852420734982484079, −8.512309075292833378548871655716, −7.26176686762151391934884976664, −6.43766743447477178824782148577, −5.53392741071857307868326735945, −4.04013264705673546255132446067, −2.68714882687767398289615949718, −0.67222527004965035599017064511, 2.94749716476945558919105274030, 3.69735046196395116160809556383, 4.59550651030658316165128463780, 5.91466025799768848820248458151, 7.45692903059643102463213334839, 8.341124383090926381563302026789, 9.216972846136810165495537243073, 10.08299783405763126390331249539, 11.05818021793447410540065309109, 12.09265320514654067907328772407

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