Properties

Degree $2$
Conductor $2352$
Sign $1$
Motivic weight $3$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 0.100·5-s + 9·9-s + 43.9·11-s + 16.6·13-s + 0.301·15-s + 121.·17-s − 127.·19-s − 53.5·23-s − 124.·25-s + 27·27-s + 235.·29-s − 18.7·31-s + 131.·33-s − 191.·37-s + 49.9·39-s + 319.·41-s + 218.·43-s + 0.904·45-s + 401.·47-s + 364.·51-s + 643.·53-s + 4.41·55-s − 381.·57-s − 11.6·59-s + 12.2·61-s + 1.67·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.00898·5-s + 0.333·9-s + 1.20·11-s + 0.355·13-s + 0.00519·15-s + 1.73·17-s − 1.53·19-s − 0.485·23-s − 0.999·25-s + 0.192·27-s + 1.50·29-s − 0.108·31-s + 0.695·33-s − 0.852·37-s + 0.205·39-s + 1.21·41-s + 0.775·43-s + 0.00299·45-s + 1.24·47-s + 1.00·51-s + 1.66·53-s + 0.0108·55-s − 0.886·57-s − 0.0256·59-s + 0.0256·61-s + 0.00319·65-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: $1$
Motivic weight: \(3\)
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.378237009\)
\(L(\frac12)\) \(\approx\) \(3.378237009\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.100T + 125T^{2} \)
11 \( 1 - 43.9T + 1.33e3T^{2} \)
13 \( 1 - 16.6T + 2.19e3T^{2} \)
17 \( 1 - 121.T + 4.91e3T^{2} \)
19 \( 1 + 127.T + 6.85e3T^{2} \)
23 \( 1 + 53.5T + 1.21e4T^{2} \)
29 \( 1 - 235.T + 2.43e4T^{2} \)
31 \( 1 + 18.7T + 2.97e4T^{2} \)
37 \( 1 + 191.T + 5.06e4T^{2} \)
41 \( 1 - 319.T + 6.89e4T^{2} \)
43 \( 1 - 218.T + 7.95e4T^{2} \)
47 \( 1 - 401.T + 1.03e5T^{2} \)
53 \( 1 - 643.T + 1.48e5T^{2} \)
59 \( 1 + 11.6T + 2.05e5T^{2} \)
61 \( 1 - 12.2T + 2.26e5T^{2} \)
67 \( 1 + 669.T + 3.00e5T^{2} \)
71 \( 1 + 822.T + 3.57e5T^{2} \)
73 \( 1 + 515.T + 3.89e5T^{2} \)
79 \( 1 - 805.T + 4.93e5T^{2} \)
83 \( 1 + 394.T + 5.71e5T^{2} \)
89 \( 1 + 673.T + 7.04e5T^{2} \)
97 \( 1 - 1.09e3T + 9.12e5T^{2} \)
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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

−8.699544707395168223402234937493, −7.928510577862917017990712081965, −7.19204988097997004476853538552, −6.25803493189634388279731340192, −5.67810439745547154991695163031, −4.30668552921410014364603549587, −3.87576287695656902629154541931, −2.83758758913221218801404914398, −1.78156884882026628038037781722, −0.832018042557373744609120251695, 0.832018042557373744609120251695, 1.78156884882026628038037781722, 2.83758758913221218801404914398, 3.87576287695656902629154541931, 4.30668552921410014364603549587, 5.67810439745547154991695163031, 6.25803493189634388279731340192, 7.19204988097997004476853538552, 7.928510577862917017990712081965, 8.699544707395168223402234937493

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