Properties

Label 2-315-63.47-c1-0-0
Degree $2$
Conductor $315$
Sign $-0.837 - 0.546i$
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  + (−0.454 + 0.262i)2-s + (0.322 − 1.70i)3-s + (−0.862 + 1.49i)4-s − 5-s + (0.299 + 0.857i)6-s + (−1.09 + 2.40i)7-s − 1.95i·8-s + (−2.79 − 1.09i)9-s + (0.454 − 0.262i)10-s − 0.685i·11-s + (2.26 + 1.94i)12-s + (−5.48 + 3.16i)13-s + (−0.134 − 1.38i)14-s + (−0.322 + 1.70i)15-s + (−1.21 − 2.10i)16-s + (1.00 + 1.74i)17-s + ⋯
L(s)  = 1  + (−0.321 + 0.185i)2-s + (0.185 − 0.982i)3-s + (−0.431 + 0.746i)4-s − 0.447·5-s + (0.122 + 0.349i)6-s + (−0.413 + 0.910i)7-s − 0.690i·8-s + (−0.930 − 0.365i)9-s + (0.143 − 0.0829i)10-s − 0.206i·11-s + (0.653 + 0.562i)12-s + (−1.52 + 0.877i)13-s + (−0.0358 − 0.369i)14-s + (−0.0831 + 0.439i)15-s + (−0.303 − 0.525i)16-s + (0.244 + 0.423i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $-0.837 - 0.546i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ -0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0780490 + 0.262238i\)
\(L(\frac12)\) \(\approx\) \(0.0780490 + 0.262238i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.322 + 1.70i)T \)
5 \( 1 + T \)
7 \( 1 + (1.09 - 2.40i)T \)
good2 \( 1 + (0.454 - 0.262i)T + (1 - 1.73i)T^{2} \)
11 \( 1 + 0.685iT - 11T^{2} \)
13 \( 1 + (5.48 - 3.16i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.00 - 1.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.11 + 2.95i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 8.06iT - 23T^{2} \)
29 \( 1 + (-0.246 - 0.142i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.98 - 2.29i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.593 - 1.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.16 + 5.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.53 - 2.65i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.57 + 9.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.01 + 4.62i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.63 + 6.30i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.70 - 0.982i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.06 - 3.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.56iT - 71T^{2} \)
73 \( 1 + (3.24 - 1.87i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.42 - 4.20i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.12 + 3.68i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (2.97 - 5.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.55 - 4.93i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.02485985524730033837616772079, −11.63468300033536429016891579496, −9.845741063860888820722171517488, −8.923313186841180707216175376064, −8.303703464893417143979896521354, −7.27676932279451982007874281660, −6.59856101679682374667022287523, −5.10954108592124093447742902877, −3.56989676008536157613005536518, −2.32232416213473275287408343931, 0.20319004113067579047937028673, 2.71832558018892273866556269255, 4.26314787922370616737929578030, 4.90032196221628584806680307540, 6.25574547273830434487270767863, 7.70098876262257963690588068562, 8.608362537964976040287379633610, 9.750529103358072512635568925990, 10.24666850105238537587018717361, 10.79329149570492244062109488061

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