Properties

Label 2-2312-1.1-c3-0-125
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  + 7.29·3-s + 8.24·5-s + 11.0·7-s + 26.2·9-s + 53.4·11-s − 0.0793·13-s + 60.1·15-s − 34.4·19-s + 80.4·21-s − 113.·23-s − 57.0·25-s − 5.52·27-s + 154.·29-s + 20.1·31-s + 389.·33-s + 90.8·35-s − 18.4·37-s − 0.578·39-s + 391.·41-s + 330.·43-s + 216.·45-s + 404.·47-s − 221.·49-s + 461.·53-s + 440.·55-s − 251.·57-s + 807.·59-s + ⋯
L(s)  = 1  + 1.40·3-s + 0.737·5-s + 0.594·7-s + 0.971·9-s + 1.46·11-s − 0.00169·13-s + 1.03·15-s − 0.416·19-s + 0.835·21-s − 1.02·23-s − 0.456·25-s − 0.0393·27-s + 0.986·29-s + 0.116·31-s + 2.05·33-s + 0.438·35-s − 0.0820·37-s − 0.00237·39-s + 1.49·41-s + 1.17·43-s + 0.716·45-s + 1.25·47-s − 0.645·49-s + 1.19·53-s + 1.07·55-s − 0.584·57-s + 1.78·59-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(5.642739884\)
\(L(\frac12)\) \(\approx\) \(5.642739884\)
\(L(\frac{5}{2})\) 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 \( 1 - 7.29T + 27T^{2} \)
5 \( 1 - 8.24T + 125T^{2} \)
7 \( 1 - 11.0T + 343T^{2} \)
11 \( 1 - 53.4T + 1.33e3T^{2} \)
13 \( 1 + 0.0793T + 2.19e3T^{2} \)
19 \( 1 + 34.4T + 6.85e3T^{2} \)
23 \( 1 + 113.T + 1.21e4T^{2} \)
29 \( 1 - 154.T + 2.43e4T^{2} \)
31 \( 1 - 20.1T + 2.97e4T^{2} \)
37 \( 1 + 18.4T + 5.06e4T^{2} \)
41 \( 1 - 391.T + 6.89e4T^{2} \)
43 \( 1 - 330.T + 7.95e4T^{2} \)
47 \( 1 - 404.T + 1.03e5T^{2} \)
53 \( 1 - 461.T + 1.48e5T^{2} \)
59 \( 1 - 807.T + 2.05e5T^{2} \)
61 \( 1 + 413.T + 2.26e5T^{2} \)
67 \( 1 - 85.0T + 3.00e5T^{2} \)
71 \( 1 - 1.07e3T + 3.57e5T^{2} \)
73 \( 1 + 282.T + 3.89e5T^{2} \)
79 \( 1 + 1.16e3T + 4.93e5T^{2} \)
83 \( 1 - 365.T + 5.71e5T^{2} \)
89 \( 1 + 274.T + 7.04e5T^{2} \)
97 \( 1 - 942.T + 9.12e5T^{2} \)
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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

−8.681782824059562539993965677125, −8.076754118706782691982830649799, −7.26338641141921927982273189457, −6.34987835831818265082735238134, −5.61317302447418623796198808029, −4.27244973287131077621961739275, −3.86427885826376491300228466816, −2.60731914127513058954565218224, −2.00154948284453380576484859892, −1.06019571479251481284949848204, 1.06019571479251481284949848204, 2.00154948284453380576484859892, 2.60731914127513058954565218224, 3.86427885826376491300228466816, 4.27244973287131077621961739275, 5.61317302447418623796198808029, 6.34987835831818265082735238134, 7.26338641141921927982273189457, 8.076754118706782691982830649799, 8.681782824059562539993965677125

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