Properties

Label 2-2352-1.1-c3-0-35
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 0.726·5-s + 9·9-s + 64.4·11-s − 71.8·13-s − 2.17·15-s − 48.9·17-s + 34.3·19-s + 0.903·23-s − 124.·25-s + 27·27-s + 226.·29-s − 275.·31-s + 193.·33-s + 295.·37-s − 215.·39-s − 186.·41-s + 455.·43-s − 6.53·45-s + 282.·47-s − 146.·51-s + 356.·53-s − 46.8·55-s + 103.·57-s + 729.·59-s − 274.·61-s + 52.1·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.0649·5-s + 0.333·9-s + 1.76·11-s − 1.53·13-s − 0.0375·15-s − 0.697·17-s + 0.415·19-s + 0.00818·23-s − 0.995·25-s + 0.192·27-s + 1.45·29-s − 1.59·31-s + 1.02·33-s + 1.31·37-s − 0.884·39-s − 0.710·41-s + 1.61·43-s − 0.0216·45-s + 0.876·47-s − 0.402·51-s + 0.923·53-s − 0.114·55-s + 0.239·57-s + 1.60·59-s − 0.575·61-s + 0.0995·65-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: \(0\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.838445650\)
\(L(\frac12)\) \(\approx\) \(2.838445650\)
\(L(\frac{5}{2})\) 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 \( 1 + 0.726T + 125T^{2} \)
11 \( 1 - 64.4T + 1.33e3T^{2} \)
13 \( 1 + 71.8T + 2.19e3T^{2} \)
17 \( 1 + 48.9T + 4.91e3T^{2} \)
19 \( 1 - 34.3T + 6.85e3T^{2} \)
23 \( 1 - 0.903T + 1.21e4T^{2} \)
29 \( 1 - 226.T + 2.43e4T^{2} \)
31 \( 1 + 275.T + 2.97e4T^{2} \)
37 \( 1 - 295.T + 5.06e4T^{2} \)
41 \( 1 + 186.T + 6.89e4T^{2} \)
43 \( 1 - 455.T + 7.95e4T^{2} \)
47 \( 1 - 282.T + 1.03e5T^{2} \)
53 \( 1 - 356.T + 1.48e5T^{2} \)
59 \( 1 - 729.T + 2.05e5T^{2} \)
61 \( 1 + 274.T + 2.26e5T^{2} \)
67 \( 1 + 193.T + 3.00e5T^{2} \)
71 \( 1 + 40.5T + 3.57e5T^{2} \)
73 \( 1 - 206.T + 3.89e5T^{2} \)
79 \( 1 + 937.T + 4.93e5T^{2} \)
83 \( 1 - 911.T + 5.71e5T^{2} \)
89 \( 1 - 949.T + 7.04e5T^{2} \)
97 \( 1 + 39.4T + 9.12e5T^{2} \)
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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

−8.824640394826639948530446460119, −7.78388801512198031749572621543, −7.15946374303968577534339029030, −6.48072152840715334481716554887, −5.47753945225793074171702242928, −4.38766330391293568108166617855, −3.89600502488203313208177934209, −2.73299720152024619801286306303, −1.90780736966263454895888517634, −0.73130621586554670798405678225, 0.73130621586554670798405678225, 1.90780736966263454895888517634, 2.73299720152024619801286306303, 3.89600502488203313208177934209, 4.38766330391293568108166617855, 5.47753945225793074171702242928, 6.48072152840715334481716554887, 7.15946374303968577534339029030, 7.78388801512198031749572621543, 8.824640394826639948530446460119

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