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Properties

Label 2-315-63.59-c1-0-24
Degree $2$
Conductor $315$
Sign $0.456 - 0.889i$
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  + (2.11 + 1.22i)2-s + (1.73 − 0.0669i)3-s + (1.98 + 3.44i)4-s − 5-s + (3.74 + 1.97i)6-s + (−2.50 − 0.855i)7-s + 4.82i·8-s + (2.99 − 0.231i)9-s + (−2.11 − 1.22i)10-s − 4.38i·11-s + (3.67 + 5.82i)12-s + (−1.54 − 0.891i)13-s + (−4.25 − 4.87i)14-s + (−1.73 + 0.0669i)15-s + (−1.92 + 3.33i)16-s + (−3.92 + 6.80i)17-s + ⋯
L(s)  = 1  + (1.49 + 0.864i)2-s + (0.999 − 0.0386i)3-s + (0.993 + 1.72i)4-s − 0.447·5-s + (1.52 + 0.805i)6-s + (−0.946 − 0.323i)7-s + 1.70i·8-s + (0.997 − 0.0772i)9-s + (−0.669 − 0.386i)10-s − 1.32i·11-s + (1.05 + 1.68i)12-s + (−0.428 − 0.247i)13-s + (−1.13 − 1.30i)14-s + (−0.446 + 0.0172i)15-s + (−0.481 + 0.833i)16-s + (−0.953 + 1.65i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.456 - 0.889i$
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: \((2,\ 315,\ (\ :1/2),\ 0.456 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.77214 + 1.69293i\)
\(L(\frac12)\) \(\approx\) \(2.77214 + 1.69293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.73 + 0.0669i)T \)
5 \( 1 + T \)
7 \( 1 + (2.50 + 0.855i)T \)
good2 \( 1 + (-2.11 - 1.22i)T + (1 + 1.73i)T^{2} \)
11 \( 1 + 4.38iT - 11T^{2} \)
13 \( 1 + (1.54 + 0.891i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.92 - 6.80i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.33 - 1.92i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 2.26iT - 23T^{2} \)
29 \( 1 + (-0.771 + 0.445i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.13 + 3.53i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.58 - 4.48i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.78 - 3.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.68 - 4.65i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.299 + 0.519i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.9 + 6.87i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.46 + 7.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.76 - 2.17i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.24 - 5.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.343iT - 71T^{2} \)
73 \( 1 + (3.38 + 1.95i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.43 - 5.95i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.09 - 5.36i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.69 - 2.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.3 + 5.95i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.46320969422923458533684083975, −11.09672523610938612793354843255, −9.935384723182136621172299035457, −8.452867965459903239045162522271, −7.969519341974464088299193613443, −6.62076413937860676584660334392, −6.17394290209356023708380413002, −4.46924724804856299601086540444, −3.71784693291991584993166520649, −2.80629362443689929967248310711, 2.23312334451951803028792419467, 2.97995909966877473023840751658, 4.23101980287652384920851762024, 4.88149442243364054613202426137, 6.58194556767852293496417618369, 7.34320692236531492082457358224, 9.018042666198041100944141654096, 9.742409367455533605573830462920, 10.73162202887906049762082123933, 11.95331629178156794852028402086

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