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Properties

Label 2-2352-7.4-c1-0-18
Degree $2$
Conductor $2352$
Sign $0.386 - 0.922i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  + (−0.5 + 0.866i)3-s + (1.5 + 2.59i)5-s + (−0.499 − 0.866i)9-s + (1.5 − 2.59i)11-s + 4·13-s − 3·15-s + (2 + 3.46i)19-s + (−2 + 3.46i)25-s + 0.999·27-s + 9·29-s + (0.5 − 0.866i)31-s + (1.5 + 2.59i)33-s + (−4 − 6.92i)37-s + (−2 + 3.46i)39-s + 10·43-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.670 + 1.16i)5-s + (−0.166 − 0.288i)9-s + (0.452 − 0.783i)11-s + 1.10·13-s − 0.774·15-s + (0.458 + 0.794i)19-s + (−0.400 + 0.692i)25-s + 0.192·27-s + 1.67·29-s + (0.0898 − 0.155i)31-s + (0.261 + 0.452i)33-s + (−0.657 − 1.13i)37-s + (−0.320 + 0.554i)39-s + 1.52·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.079128615\)
\(L(\frac12)\) \(\approx\) \(2.079128615\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−9.191194457034172033667068140285, −8.493726130662376702176371268737, −7.52922502713470714085594981468, −6.49577817303767943700305446714, −6.12314901732677263346978377256, −5.41334780524496250871749050665, −4.13066190862456696357548644828, −3.40681126651736352342920551504, −2.55037765749452042703820773053, −1.14053937374598187484560860399, 0.934623961356220186644608786142, 1.62703279519829525825126635830, 2.85416877116906002063342834646, 4.23180050874328566971402585347, 4.92037081186123766436552555962, 5.71224896492794934933493189596, 6.47962039790291070061279743683, 7.18695714719766683317517965209, 8.244285315937869529335490538634, 8.822081953423067136385599064625

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