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Properties

Label 2-315-63.59-c1-0-19
Degree $2$
Conductor $315$
Sign $0.192 + 0.981i$
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  + (0.794 + 0.458i)2-s + (−1.63 + 0.572i)3-s + (−0.578 − 1.00i)4-s + 5-s + (−1.56 − 0.295i)6-s + (−2.55 − 0.672i)7-s − 2.89i·8-s + (2.34 − 1.87i)9-s + (0.794 + 0.458i)10-s − 3.32i·11-s + (1.51 + 1.30i)12-s + (−2.08 − 1.20i)13-s + (−1.72 − 1.70i)14-s + (−1.63 + 0.572i)15-s + (0.172 − 0.298i)16-s + (−0.514 + 0.890i)17-s + ⋯
L(s)  = 1  + (0.562 + 0.324i)2-s + (−0.943 + 0.330i)3-s + (−0.289 − 0.501i)4-s + 0.447·5-s + (−0.637 − 0.120i)6-s + (−0.967 − 0.254i)7-s − 1.02i·8-s + (0.781 − 0.623i)9-s + (0.251 + 0.145i)10-s − 1.00i·11-s + (0.438 + 0.377i)12-s + (−0.578 − 0.333i)13-s + (−0.461 − 0.456i)14-s + (−0.422 + 0.147i)15-s + (0.0431 − 0.0747i)16-s + (−0.124 + 0.216i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.192 + 0.981i$
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: \((2,\ 315,\ (\ :1/2),\ 0.192 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.718156 - 0.590764i\)
\(L(\frac12)\) \(\approx\) \(0.718156 - 0.590764i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.63 - 0.572i)T \)
5 \( 1 - T \)
7 \( 1 + (2.55 + 0.672i)T \)
good2 \( 1 + (-0.794 - 0.458i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + 3.32iT - 11T^{2} \)
13 \( 1 + (2.08 + 1.20i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.514 - 0.890i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.75 + 2.74i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 7.46iT - 23T^{2} \)
29 \( 1 + (3.78 - 2.18i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.455 + 0.262i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.83 - 6.64i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0682 - 0.118i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.65 + 9.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.51 - 7.81i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.24 - 0.716i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.319 + 0.553i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-8.85 - 5.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.02 - 3.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.03iT - 71T^{2} \)
73 \( 1 + (-5.39 - 3.11i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.27 + 2.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.30 + 3.99i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.33 + 12.7i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.8 + 8.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.41762798973752118860102389713, −10.36465104712676856326411772208, −9.872204482019673902053725288790, −8.907860483604775311407607568544, −7.06624491671085989867289684407, −6.29194025895844624091567791310, −5.54446156531214515283554477222, −4.60910931985770580481988476726, −3.29906715134298130077907572474, −0.63339738863484123717149727001, 2.11076763551975276546361022294, 3.63101969656234331788503890185, 4.93797557237813250824276842281, 5.72106360340392528174271743632, 6.95742763312614400552842734933, 7.77188229356774158082586076066, 9.478228953069133447553688277738, 9.870038567752811695768603951151, 11.37071070121634802728069204898, 11.95010755799910308531720559388

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