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Properties

Label 2-2352-12.11-c1-0-55
Degree $2$
Conductor $2352$
Sign $0.957 - 0.288i$
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  + (1.65 − 0.5i)3-s + 3.31i·5-s + (2.5 − 1.65i)9-s + 3.31·11-s + 4·13-s + (1.65 + 5.5i)15-s − 3.31i·17-s − 7i·19-s + 3.31·23-s − 6·25-s + (3.31 − 4i)27-s + 6.63i·29-s − 3i·31-s + (5.5 − 1.65i)33-s − 37-s + ⋯
L(s)  = 1  + (0.957 − 0.288i)3-s + 1.48i·5-s + (0.833 − 0.552i)9-s + 1.00·11-s + 1.10·13-s + (0.428 + 1.42i)15-s − 0.804i·17-s − 1.60i·19-s + 0.691·23-s − 1.20·25-s + (0.638 − 0.769i)27-s + 1.23i·29-s − 0.538i·31-s + (0.957 − 0.288i)33-s − 0.164·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.957 - 0.288i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.957 - 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.999035660\)
\(L(\frac12)\) \(\approx\) \(2.999035660\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 + 0.5i)T \)
7 \( 1 \)
good5 \( 1 - 3.31iT - 5T^{2} \)
11 \( 1 - 3.31T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + 3.31iT - 17T^{2} \)
19 \( 1 + 7iT - 19T^{2} \)
23 \( 1 - 3.31T + 23T^{2} \)
29 \( 1 - 6.63iT - 29T^{2} \)
31 \( 1 + 3iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 - 6.63iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 9.94T + 47T^{2} \)
53 \( 1 + 3.31iT - 53T^{2} \)
59 \( 1 - 3.31T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 - 9iT - 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 - 9iT - 79T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + 3.31iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.055905158937746444551401469835, −8.292964245116075479904431005609, −7.28989583011366000756979781830, −6.77776800056090715414529370585, −6.38138251939944851980219349958, −4.95205519805719681484466041083, −3.80365606417093725535866851125, −3.17337151091118136168354972091, −2.46671361603398062678100262791, −1.19423749366218509530302355984, 1.20762295331120068890166440811, 1.83974472691206594939570096136, 3.48877681944885302994491703911, 3.97147758249226579107430812738, 4.77913634018938331496041938545, 5.76940372992010244081191866557, 6.57055562819779860949080263308, 7.84056297583027880347191668779, 8.307673676515206540610236705163, 8.890617003677781350780972036518

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