Properties

Label 2-2352-1.1-c3-0-73
Degree $2$
Conductor $2352$
Sign $-1$
Analytic cond. $138.772$
Root an. cond. $11.7801$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 20.8·5-s + 9·9-s − 15.1·11-s − 2.16·13-s − 62.5·15-s + 119.·17-s − 33.5·19-s − 0.651·23-s + 309.·25-s + 27·27-s − 163.·29-s − 223.·31-s − 45.4·33-s + 168.·37-s − 6.48·39-s + 323.·41-s − 221.·43-s − 187.·45-s + 508.·47-s + 358.·51-s − 176.·53-s + 315.·55-s − 100.·57-s + 454.·59-s − 38.6·61-s + 45.0·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.86·5-s + 0.333·9-s − 0.415·11-s − 0.0461·13-s − 1.07·15-s + 1.70·17-s − 0.404·19-s − 0.00590·23-s + 2.47·25-s + 0.192·27-s − 1.04·29-s − 1.29·31-s − 0.239·33-s + 0.748·37-s − 0.0266·39-s + 1.23·41-s − 0.785·43-s − 0.621·45-s + 1.57·47-s + 0.983·51-s − 0.457·53-s + 0.774·55-s − 0.233·57-s + 1.00·59-s − 0.0811·61-s + 0.0860·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(138.772\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
7 \( 1 \)
good5 \( 1 + 20.8T + 125T^{2} \)
11 \( 1 + 15.1T + 1.33e3T^{2} \)
13 \( 1 + 2.16T + 2.19e3T^{2} \)
17 \( 1 - 119.T + 4.91e3T^{2} \)
19 \( 1 + 33.5T + 6.85e3T^{2} \)
23 \( 1 + 0.651T + 1.21e4T^{2} \)
29 \( 1 + 163.T + 2.43e4T^{2} \)
31 \( 1 + 223.T + 2.97e4T^{2} \)
37 \( 1 - 168.T + 5.06e4T^{2} \)
41 \( 1 - 323.T + 6.89e4T^{2} \)
43 \( 1 + 221.T + 7.95e4T^{2} \)
47 \( 1 - 508.T + 1.03e5T^{2} \)
53 \( 1 + 176.T + 1.48e5T^{2} \)
59 \( 1 - 454.T + 2.05e5T^{2} \)
61 \( 1 + 38.6T + 2.26e5T^{2} \)
67 \( 1 + 141.T + 3.00e5T^{2} \)
71 \( 1 + 602.T + 3.57e5T^{2} \)
73 \( 1 - 1.10e3T + 3.89e5T^{2} \)
79 \( 1 - 116.T + 4.93e5T^{2} \)
83 \( 1 + 568.T + 5.71e5T^{2} \)
89 \( 1 - 383.T + 7.04e5T^{2} \)
97 \( 1 + 334.T + 9.12e5T^{2} \)
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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

−8.078438837358060382094484112350, −7.58193314722444538517062981921, −7.15872249860703764316768439876, −5.84051949655477768653289217985, −4.90505587889398848558486199389, −3.90868692286935363576877822366, −3.52881319785242172628897112093, −2.52984359047893878765173537965, −1.07587806306217472055390805781, 0, 1.07587806306217472055390805781, 2.52984359047893878765173537965, 3.52881319785242172628897112093, 3.90868692286935363576877822366, 4.90505587889398848558486199389, 5.84051949655477768653289217985, 7.15872249860703764316768439876, 7.58193314722444538517062981921, 8.078438837358060382094484112350

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