↑

Properties

Label 2-756-12.11-c1-0-8
Degree $2$
Conductor $756$
Sign $-0.991 - 0.130i$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
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  + (0.932 + 1.06i)2-s + (−0.260 + 1.98i)4-s + 0.546i·5-s + i·7-s + (−2.35 + 1.57i)8-s + (−0.580 + 0.509i)10-s − 4.32·11-s − 3.39·13-s + (−1.06 + 0.932i)14-s + (−3.86 − 1.03i)16-s + 0.0960i·17-s + 7.78i·19-s + (−1.08 − 0.142i)20-s + (−4.03 − 4.60i)22-s + 0.716·23-s + ⋯
L(s)  = 1  + (0.659 + 0.751i)2-s + (−0.130 + 0.991i)4-s + 0.244i·5-s + 0.377i·7-s + (−0.831 + 0.555i)8-s + (−0.183 + 0.161i)10-s − 1.30·11-s − 0.940·13-s + (−0.284 + 0.249i)14-s + (−0.965 − 0.258i)16-s + 0.0232i·17-s + 1.78i·19-s + (−0.242 − 0.0318i)20-s + (−0.860 − 0.981i)22-s + 0.149·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.991 - 0.130i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0916988 + 1.40018i\)
\(L(\frac12)\) \(\approx\) \(0.0916988 + 1.40018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.932 - 1.06i)T \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 - 0.546iT - 5T^{2} \)
11 \( 1 + 4.32T + 11T^{2} \)
13 \( 1 + 3.39T + 13T^{2} \)
17 \( 1 - 0.0960iT - 17T^{2} \)
19 \( 1 - 7.78iT - 19T^{2} \)
23 \( 1 - 0.716T + 23T^{2} \)
29 \( 1 - 2.80iT - 29T^{2} \)
31 \( 1 - 1.38iT - 31T^{2} \)
37 \( 1 - 9.87T + 37T^{2} \)
41 \( 1 + 9.42iT - 41T^{2} \)
43 \( 1 - 1.12iT - 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 10.3iT - 53T^{2} \)
59 \( 1 + 7.15T + 59T^{2} \)
61 \( 1 - 4.47T + 61T^{2} \)
67 \( 1 - 10.1iT - 67T^{2} \)
71 \( 1 - 5.68T + 71T^{2} \)
73 \( 1 - 0.760T + 73T^{2} \)
79 \( 1 + 1.95iT - 79T^{2} \)
83 \( 1 - 3.56T + 83T^{2} \)
89 \( 1 + 5.30iT - 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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

−10.73952491578626081190900738028, −9.903905460165013831267939378345, −8.808171374174978696062369161339, −7.903559528269902307871476471875, −7.35915218178478398143901396459, −6.23645358359486772146906480975, −5.42710682538067558418441544403, −4.64709347127646489791721967332, −3.34408028887487281353303671128, −2.39136979511268137023548090407, 0.54324513795519620857950958087, 2.32929221027912696342469990815, 3.14021382599380216558402943251, 4.77415736010219679780752236148, 4.87399284997690887366069078043, 6.25346358594410821097604695135, 7.23492516033854538896435760622, 8.267296953037790704872112567909, 9.460786725024985320775392790023, 9.977981301678917623396995855042

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