↑

Properties

Label 2-912-57.56-c1-0-32
Degree $2$
Conductor $912$
Sign $0.187 + 0.982i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.58 − 0.707i)3-s − 2.23i·5-s + 7-s + (2.00 − 2.23i)9-s + 2.23i·11-s − 4.24i·13-s + (−1.58 − 3.53i)15-s − 2.23i·17-s + (−1 + 4.24i)19-s + (1.58 − 0.707i)21-s − 4.47i·23-s + (1.58 − 4.94i)27-s + 4.24i·31-s + (1.58 + 3.53i)33-s − 2.23i·35-s + ⋯
L(s)  = 1  + (0.912 − 0.408i)3-s − 0.999i·5-s + 0.377·7-s + (0.666 − 0.745i)9-s + 0.674i·11-s − 1.17i·13-s + (−0.408 − 0.912i)15-s − 0.542i·17-s + (−0.229 + 0.973i)19-s + (0.345 − 0.154i)21-s − 0.932i·23-s + (0.304 − 0.952i)27-s + 0.762i·31-s + (0.275 + 0.615i)33-s − 0.377i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.187 + 0.982i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.187 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70509 - 1.40976i\)
\(L(\frac12)\) \(\approx\) \(1.70509 - 1.40976i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 + 0.707i)T \)
19 \( 1 + (1 - 4.24i)T \)
good5 \( 1 + 2.23iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 - 2.23iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 2.23iT - 17T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 4.24iT - 31T^{2} \)
37 \( 1 - 4.24iT - 37T^{2} \)
41 \( 1 + 9.48T + 41T^{2} \)
43 \( 1 - 7T + 43T^{2} \)
47 \( 1 + 11.1iT - 47T^{2} \)
53 \( 1 - 9.48T + 53T^{2} \)
59 \( 1 + 9.48T + 59T^{2} \)
61 \( 1 - 11T + 61T^{2} \)
67 \( 1 - 8.48iT - 67T^{2} \)
71 \( 1 + 9.48T + 71T^{2} \)
73 \( 1 - 5T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 8.94iT - 83T^{2} \)
89 \( 1 - 9.48T + 89T^{2} \)
97 \( 1 - 12.7iT - 97T^{2} \)
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

−9.887477198973277570823074447447, −8.761228097552828982350520433470, −8.389561039699833366494642983955, −7.57405265101439439575864687126, −6.66127639377718116589764815066, −5.37269660583683378031665794278, −4.56716765153207412389039567749, −3.43814795633499829304597429383, −2.20969593180293654529533231048, −1.01506031452162068267708858586, 1.86445880725668964911688173518, 2.91210354417246008104177653987, 3.82617426319607571130913320025, 4.76324582709315053928655080819, 6.08056555411737259326830452037, 7.02997054001974366334662321044, 7.74373611913859379478461744567, 8.740728913634239507167695695258, 9.324112042883773172234390105923, 10.27531079369628560329089478507

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