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Properties

Label 2-2352-21.20-c1-0-24
Degree $2$
Conductor $2352$
Sign $-0.163 - 0.986i$
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  + (1.33 + 1.10i)3-s − 0.145·5-s + (0.554 + 2.94i)9-s − 2.46i·11-s − 2.04i·13-s + (−0.193 − 0.160i)15-s − 1.75·17-s + 4.25i·19-s + 8.61i·23-s − 4.97·25-s + (−2.52 + 4.54i)27-s + 7.08i·29-s + 3.60i·31-s + (2.73 − 3.29i)33-s + 5.86·37-s + ⋯
L(s)  = 1  + (0.769 + 0.638i)3-s − 0.0649·5-s + (0.184 + 0.982i)9-s − 0.744i·11-s − 0.566i·13-s + (−0.0500 − 0.0414i)15-s − 0.426·17-s + 0.975i·19-s + 1.79i·23-s − 0.995·25-s + (−0.485 + 0.874i)27-s + 1.31i·29-s + 0.646i·31-s + (0.475 − 0.573i)33-s + 0.964·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.163 - 0.986i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.163 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.987333694\)
\(L(\frac12)\) \(\approx\) \(1.987333694\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.33 - 1.10i)T \)
7 \( 1 \)
good5 \( 1 + 0.145T + 5T^{2} \)
11 \( 1 + 2.46iT - 11T^{2} \)
13 \( 1 + 2.04iT - 13T^{2} \)
17 \( 1 + 1.75T + 17T^{2} \)
19 \( 1 - 4.25iT - 19T^{2} \)
23 \( 1 - 8.61iT - 23T^{2} \)
29 \( 1 - 7.08iT - 29T^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 - 5.86T + 37T^{2} \)
41 \( 1 + 5.33T + 41T^{2} \)
43 \( 1 - 9.19T + 43T^{2} \)
47 \( 1 - 9.30T + 47T^{2} \)
53 \( 1 - 5.19iT - 53T^{2} \)
59 \( 1 - 11.2T + 59T^{2} \)
61 \( 1 - 5.38iT - 61T^{2} \)
67 \( 1 - 5.14T + 67T^{2} \)
71 \( 1 - 7.79iT - 71T^{2} \)
73 \( 1 + 13.0iT - 73T^{2} \)
79 \( 1 - 5.72T + 79T^{2} \)
83 \( 1 + 15.9T + 83T^{2} \)
89 \( 1 + 8.68T + 89T^{2} \)
97 \( 1 - 6.65iT - 97T^{2} \)
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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

−9.171992266591705600896654038933, −8.473243069792075183987490103503, −7.80164864391458321537087958349, −7.12875745161252868423234194119, −5.79476531689447949359705252967, −5.35996774371883101620670715786, −4.12297872040689934896327683552, −3.53670320527824695897368922720, −2.66465435023238375513741722367, −1.42816561843174125948019718418, 0.61023505628231267208804643012, 2.15727162772985858481669936498, 2.55490885913300834813240229386, 4.02453282134706140986952884298, 4.45939759770794094711380992115, 5.83353812458976452304373599276, 6.67497501498451750299201813927, 7.19359954980732890432330770996, 8.063085097971496380158617097382, 8.661864567284992266976767988954

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