↑

Properties

Label 2-2312-136.133-c0-0-0
Degree $2$
Conductor $2312$
Sign $0.547 + 0.837i$
Analytic cond. $1.15383$
Root an. cond. $1.07416$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)2-s + (−1.38 − 0.275i)3-s + (0.707 − 0.707i)4-s + (0.785 + 1.17i)5-s + (−1.38 + 0.275i)6-s + (0.382 − 0.923i)8-s + (0.923 + 0.382i)9-s + (1.17 + 0.785i)10-s + (−0.275 − 1.38i)11-s + (−1.17 + 0.785i)12-s + (−0.765 − 1.84i)15-s − i·16-s + 1.00·18-s + (1.38 + 0.275i)20-s + (−0.785 − 1.17i)22-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (−1.38 − 0.275i)3-s + (0.707 − 0.707i)4-s + (0.785 + 1.17i)5-s + (−1.38 + 0.275i)6-s + (0.382 − 0.923i)8-s + (0.923 + 0.382i)9-s + (1.17 + 0.785i)10-s + (−0.275 − 1.38i)11-s + (−1.17 + 0.785i)12-s + (−0.765 − 1.84i)15-s − i·16-s + 1.00·18-s + (1.38 + 0.275i)20-s + (−0.785 − 1.17i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.547 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2312\)    =    \(2^{3} \cdot 17^{2}\)
Sign: $0.547 + 0.837i$
Analytic conductor: \(1.15383\)
Root analytic conductor: \(1.07416\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2312} (1221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2312,\ (\ :0),\ 0.547 + 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.520111881\)
\(L(\frac12)\) \(\approx\) \(1.520111881\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
17 \( 1 \)
good3 \( 1 + (1.38 + 0.275i)T + (0.923 + 0.382i)T^{2} \)
5 \( 1 + (-0.785 - 1.17i)T + (-0.382 + 0.923i)T^{2} \)
7 \( 1 + (0.382 + 0.923i)T^{2} \)
11 \( 1 + (0.275 + 1.38i)T + (-0.923 + 0.382i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.923 + 0.382i)T^{2} \)
29 \( 1 + (-1.17 + 0.785i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (-1.38 - 0.275i)T + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
53 \( 1 + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (1.17 + 0.785i)T + (0.382 + 0.923i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (0.382 - 0.923i)T^{2} \)
79 \( 1 + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
97 \( 1 + (-0.382 + 0.923i)T^{2} \)
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

−9.454760604998067266137052746162, −8.061048267830700623960537375438, −7.02409678212267649719842118029, −6.28363936886311859290776790836, −6.04564996988760530947995011520, −5.35679382313591752778950869288, −4.37236869105741978521417461022, −3.15184155752647877895460746687, −2.47131333488509343314237022326, −1.03142506491805876867180302776, 1.40955993954904325434531075136, 2.59885851929045114096261294468, 4.26749140761956420035970812603, 4.68862143437233661927470517832, 5.35160619479713017174257202848, 5.89273273849261947512082712250, 6.69486360510665435155972290842, 7.47145072533523048041440103431, 8.495956411575299680334121017521, 9.388882962092775753454889595596

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