↑

Properties

Label 2-756-84.23-c1-0-50
Degree $2$
Conductor $756$
Sign $-0.877 - 0.479i$
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  + (−0.498 − 1.32i)2-s + (−1.50 + 1.31i)4-s + (−0.936 − 0.540i)5-s + (0.749 + 2.53i)7-s + (2.49 + 1.33i)8-s + (−0.249 + 1.50i)10-s + (−2.43 − 4.22i)11-s + 0.815·13-s + (2.98 − 2.25i)14-s + (0.521 − 3.96i)16-s + (−1.47 + 0.848i)17-s + (−3.58 − 2.07i)19-s + (2.12 − 0.421i)20-s + (−4.37 + 5.33i)22-s + (−1.75 + 3.04i)23-s + ⋯
L(s)  = 1  + (−0.352 − 0.935i)2-s + (−0.751 + 0.659i)4-s + (−0.418 − 0.241i)5-s + (0.283 + 0.959i)7-s + (0.881 + 0.471i)8-s + (−0.0787 + 0.477i)10-s + (−0.735 − 1.27i)11-s + 0.226·13-s + (0.797 − 0.602i)14-s + (0.130 − 0.991i)16-s + (−0.356 + 0.205i)17-s + (−0.823 − 0.475i)19-s + (0.474 − 0.0943i)20-s + (−0.933 + 1.13i)22-s + (−0.366 + 0.634i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.877 - 0.479i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.877 - 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0600686 + 0.235102i\)
\(L(\frac12)\) \(\approx\) \(0.0600686 + 0.235102i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.498 + 1.32i)T \)
3 \( 1 \)
7 \( 1 + (-0.749 - 2.53i)T \)
good5 \( 1 + (0.936 + 0.540i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.43 + 4.22i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.815T + 13T^{2} \)
17 \( 1 + (1.47 - 0.848i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.58 + 2.07i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.75 - 3.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9.61iT - 29T^{2} \)
31 \( 1 + (7.73 - 4.46i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.37 - 5.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.96iT - 41T^{2} \)
43 \( 1 - 0.510iT - 43T^{2} \)
47 \( 1 + (3.40 - 5.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.99 - 1.73i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.50 - 4.34i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.85 + 11.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.66 + 2.11i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + (1.49 + 2.59i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.41 + 1.97i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 + (-0.313 - 0.180i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.12T + 97T^{2} \)
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

−9.917556791347434686906100977906, −8.850708472701274822623345690703, −8.434571958443969025475729726110, −7.70283623166260724002600121499, −6.14162231458395966786864043517, −5.19642456705844290065697409392, −4.10967707467306622996518391925, −3.02867458059843658932717303032, −1.95659571768184217199828226439, −0.13511429180348899391085543387, 1.79596850565500021767610879248, 3.78689760095832167814806787369, 4.59662991926669110464560554873, 5.54864706841942178699223961867, 6.86847719281767837695060527940, 7.28718710531547170953824357992, 8.063837757314434372511631871009, 8.974241722307118475654119670349, 10.04569773932572928564275122308, 10.54488885776662459861009517020

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