Properties

Label 2-2312-1.1-c3-0-51
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.95·3-s + 4.89·5-s + 4.46·7-s − 18.2·9-s − 60.3·11-s − 56.1·13-s + 14.4·15-s + 134.·19-s + 13.1·21-s − 39.1·23-s − 101.·25-s − 133.·27-s − 113.·29-s + 306.·31-s − 178.·33-s + 21.8·35-s − 61.9·37-s − 165.·39-s + 317.·41-s − 122.·43-s − 89.4·45-s + 303.·47-s − 323.·49-s − 133.·53-s − 295.·55-s + 398.·57-s + 130.·59-s + ⋯
L(s)  = 1  + 0.568·3-s + 0.437·5-s + 0.240·7-s − 0.677·9-s − 1.65·11-s − 1.19·13-s + 0.248·15-s + 1.62·19-s + 0.136·21-s − 0.355·23-s − 0.808·25-s − 0.953·27-s − 0.728·29-s + 1.77·31-s − 0.940·33-s + 0.105·35-s − 0.275·37-s − 0.681·39-s + 1.20·41-s − 0.435·43-s − 0.296·45-s + 0.941·47-s − 0.941·49-s − 0.346·53-s − 0.724·55-s + 0.925·57-s + 0.288·59-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

\(L(2)\) \(\approx\) \(2.024933471\)
\(L(\frac12)\) \(\approx\) \(2.024933471\)
\(L(\frac{5}{2})\) 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 \( 1 - 2.95T + 27T^{2} \)
5 \( 1 - 4.89T + 125T^{2} \)
7 \( 1 - 4.46T + 343T^{2} \)
11 \( 1 + 60.3T + 1.33e3T^{2} \)
13 \( 1 + 56.1T + 2.19e3T^{2} \)
19 \( 1 - 134.T + 6.85e3T^{2} \)
23 \( 1 + 39.1T + 1.21e4T^{2} \)
29 \( 1 + 113.T + 2.43e4T^{2} \)
31 \( 1 - 306.T + 2.97e4T^{2} \)
37 \( 1 + 61.9T + 5.06e4T^{2} \)
41 \( 1 - 317.T + 6.89e4T^{2} \)
43 \( 1 + 122.T + 7.95e4T^{2} \)
47 \( 1 - 303.T + 1.03e5T^{2} \)
53 \( 1 + 133.T + 1.48e5T^{2} \)
59 \( 1 - 130.T + 2.05e5T^{2} \)
61 \( 1 - 772.T + 2.26e5T^{2} \)
67 \( 1 - 378.T + 3.00e5T^{2} \)
71 \( 1 + 465.T + 3.57e5T^{2} \)
73 \( 1 - 664.T + 3.89e5T^{2} \)
79 \( 1 - 925.T + 4.93e5T^{2} \)
83 \( 1 - 723.T + 5.71e5T^{2} \)
89 \( 1 - 889.T + 7.04e5T^{2} \)
97 \( 1 - 1.50e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.545675851106446384501263592647, −7.73942934434945086544255652053, −7.54148477141234777389364587529, −6.20057792181921252510405354970, −5.34487038490503130017985647900, −4.92730042227769573635495752005, −3.55560502281529032619178179268, −2.63171556186272312869132850600, −2.17020366680112219806511484512, −0.58332848716255313965586445695, 0.58332848716255313965586445695, 2.17020366680112219806511484512, 2.63171556186272312869132850600, 3.55560502281529032619178179268, 4.92730042227769573635495752005, 5.34487038490503130017985647900, 6.20057792181921252510405354970, 7.54148477141234777389364587529, 7.73942934434945086544255652053, 8.545675851106446384501263592647

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