Properties

Label 2-2312-1.1-c3-0-26
Degree $2$
Conductor $2312$
Sign $1$
Analytic cond. $136.412$
Root an. cond. $11.6795$
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  − 8.52·3-s − 18.2·5-s + 13.8·7-s + 45.6·9-s − 60.8·11-s + 61.8·13-s + 155.·15-s + 40.0·19-s − 118.·21-s − 4.88·23-s + 208.·25-s − 158.·27-s − 113.·29-s − 95.1·31-s + 518.·33-s − 253.·35-s + 273.·37-s − 527.·39-s + 446.·41-s + 274.·43-s − 834.·45-s − 27.6·47-s − 150.·49-s − 488.·53-s + 1.11e3·55-s − 341.·57-s − 266.·59-s + ⋯
L(s)  = 1  − 1.64·3-s − 1.63·5-s + 0.749·7-s + 1.69·9-s − 1.66·11-s + 1.31·13-s + 2.68·15-s + 0.483·19-s − 1.22·21-s − 0.0443·23-s + 1.67·25-s − 1.13·27-s − 0.724·29-s − 0.551·31-s + 2.73·33-s − 1.22·35-s + 1.21·37-s − 2.16·39-s + 1.69·41-s + 0.972·43-s − 2.76·45-s − 0.0858·47-s − 0.438·49-s − 1.26·53-s + 2.72·55-s − 0.792·57-s − 0.587·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(136.412\)
Root analytic conductor: \(11.6795\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4685545571\)
\(L(\frac12)\) \(\approx\) \(0.4685545571\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + 8.52T + 27T^{2} \)
5 \( 1 + 18.2T + 125T^{2} \)
7 \( 1 - 13.8T + 343T^{2} \)
11 \( 1 + 60.8T + 1.33e3T^{2} \)
13 \( 1 - 61.8T + 2.19e3T^{2} \)
19 \( 1 - 40.0T + 6.85e3T^{2} \)
23 \( 1 + 4.88T + 1.21e4T^{2} \)
29 \( 1 + 113.T + 2.43e4T^{2} \)
31 \( 1 + 95.1T + 2.97e4T^{2} \)
37 \( 1 - 273.T + 5.06e4T^{2} \)
41 \( 1 - 446.T + 6.89e4T^{2} \)
43 \( 1 - 274.T + 7.95e4T^{2} \)
47 \( 1 + 27.6T + 1.03e5T^{2} \)
53 \( 1 + 488.T + 1.48e5T^{2} \)
59 \( 1 + 266.T + 2.05e5T^{2} \)
61 \( 1 + 502.T + 2.26e5T^{2} \)
67 \( 1 + 1.00e3T + 3.00e5T^{2} \)
71 \( 1 - 724.T + 3.57e5T^{2} \)
73 \( 1 + 188.T + 3.89e5T^{2} \)
79 \( 1 - 48.3T + 4.93e5T^{2} \)
83 \( 1 + 1.38e3T + 5.71e5T^{2} \)
89 \( 1 + 50.3T + 7.04e5T^{2} \)
97 \( 1 + 1.28e3T + 9.12e5T^{2} \)
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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

−8.320880516252291834641439226353, −7.73327236474862539326636890507, −7.30011447825187772697079686261, −6.09499392773506972578796686049, −5.53110193544071531983203531037, −4.65688392895590818730711702604, −4.13767199035217215179796526018, −2.98543286307463978754164145891, −1.32226130048417369280591007274, −0.37126299259068236974528608049, 0.37126299259068236974528608049, 1.32226130048417369280591007274, 2.98543286307463978754164145891, 4.13767199035217215179796526018, 4.65688392895590818730711702604, 5.53110193544071531983203531037, 6.09499392773506972578796686049, 7.30011447825187772697079686261, 7.73327236474862539326636890507, 8.320880516252291834641439226353

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