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Properties

Label 2-2352-7.6-c2-0-5
Degree $2$
Conductor $2352$
Sign $0.654 - 0.755i$
Analytic cond. $64.0873$
Root an. cond. $8.00545$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.73i·3-s − 9.58i·5-s − 2.99·9-s − 4.59·11-s − 7.84i·13-s − 16.5·15-s + 19.1i·17-s + 16.4i·19-s + 24·23-s − 66.7·25-s + 5.19i·27-s − 10.5·29-s + 55.7i·31-s + 7.95i·33-s − 24.4·37-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.91i·5-s − 0.333·9-s − 0.417·11-s − 0.603i·13-s − 1.10·15-s + 1.12i·17-s + 0.863i·19-s + 1.04·23-s − 2.67·25-s + 0.192i·27-s − 0.365·29-s + 1.79i·31-s + 0.241i·33-s − 0.659·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.654 - 0.755i$
Analytic conductor: \(64.0873\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1),\ 0.654 - 0.755i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6597446279\)
\(L(\frac12)\) \(\approx\) \(0.6597446279\)
\(L(2)\) 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 \( 1 + 1.73iT \)
7 \( 1 \)
good5 \( 1 + 9.58iT - 25T^{2} \)
11 \( 1 + 4.59T + 121T^{2} \)
13 \( 1 + 7.84iT - 169T^{2} \)
17 \( 1 - 19.1iT - 289T^{2} \)
19 \( 1 - 16.4iT - 361T^{2} \)
23 \( 1 - 24T + 529T^{2} \)
29 \( 1 + 10.5T + 841T^{2} \)
31 \( 1 - 55.7iT - 961T^{2} \)
37 \( 1 + 24.4T + 1.36e3T^{2} \)
41 \( 1 - 48.7iT - 1.68e3T^{2} \)
43 \( 1 - 18.7T + 1.84e3T^{2} \)
47 \( 1 - 12.0iT - 2.20e3T^{2} \)
53 \( 1 + 37.4T + 2.80e3T^{2} \)
59 \( 1 - 63.1iT - 3.48e3T^{2} \)
61 \( 1 + 111. iT - 3.72e3T^{2} \)
67 \( 1 + 63.9T + 4.48e3T^{2} \)
71 \( 1 + 21.5T + 5.04e3T^{2} \)
73 \( 1 - 61.8iT - 5.32e3T^{2} \)
79 \( 1 - 3.81T + 6.24e3T^{2} \)
83 \( 1 - 100. iT - 6.88e3T^{2} \)
89 \( 1 - 99.0iT - 7.92e3T^{2} \)
97 \( 1 + 63.5iT - 9.40e3T^{2} \)
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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

−8.702154205433995046874753467368, −8.227325474416232428180647197010, −7.65072422294699993751742000837, −6.51234886316161201164051306877, −5.58330330054334043559700525707, −5.12369281884926042446297686524, −4.20283993645665554190671858539, −3.13525470112989965720783534382, −1.68069253572946154201350198612, −1.11155004996559074967842718405, 0.16643595394940157455100049632, 2.23609760803049206521442804129, 2.86346045338314995608482051424, 3.66207164186192474541234203791, 4.64040998926548491227648035484, 5.62337086005818864775771466617, 6.46169202032146553716569566593, 7.22403871747885071355557038790, 7.60560645949094790761311573931, 8.943375186765347219913997865308

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