↑

Properties

Label 2-756-21.17-c1-0-2
Degree $2$
Conductor $756$
Sign $-0.444 - 0.895i$
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  + (2.12 + 3.67i)5-s + (−2.5 + 0.866i)7-s + 3.46i·13-s + (2.12 − 3.67i)17-s + (−1.5 + 0.866i)19-s + (−6.36 + 3.67i)23-s + (−6.5 + 11.2i)25-s − 7.34i·29-s + (−1.5 − 0.866i)31-s + (−8.48 − 7.34i)35-s + (4 + 6.92i)37-s + 4.24·41-s − 5·43-s + (2.12 + 3.67i)47-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (0.948 + 1.64i)5-s + (−0.944 + 0.327i)7-s + 0.960i·13-s + (0.514 − 0.891i)17-s + (−0.344 + 0.198i)19-s + (−1.32 + 0.766i)23-s + (−1.30 + 2.25i)25-s − 1.36i·29-s + (−0.269 − 0.155i)31-s + (−1.43 − 1.24i)35-s + (0.657 + 1.13i)37-s + 0.662·41-s − 0.762·43-s + (0.309 + 0.535i)47-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.444 - 0.895i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -0.444 - 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.719884 + 1.16035i\)
\(L(\frac12)\) \(\approx\) \(0.719884 + 1.16035i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (-2.12 - 3.67i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (-2.12 + 3.67i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.36 - 3.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.34iT - 29T^{2} \)
31 \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.24T + 41T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + (-2.12 - 3.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.36 - 3.67i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.24 + 7.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.5 - 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.6iT - 71T^{2} \)
73 \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + (-6.36 - 11.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.5iT - 97T^{2} \)
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

−10.40882781759027092870249275463, −9.737629502249309594790110731946, −9.369742857881156901017536826627, −7.85509721703072805242495149088, −6.92803538710350401018622812709, −6.27408344672782433877591988826, −5.65014448385017786700796988191, −3.97193173549699606501310061513, −2.91784594219035810158006309390, −2.09889685984103982257925882341, 0.67267518439692856126443497282, 2.05993082235408947955660919137, 3.60463448874715940383023628412, 4.69111130349796032932759645256, 5.71227122782334531677062808648, 6.21635387985648609658430603056, 7.61020688176111991164443100135, 8.583159608735999818775298076181, 9.140226220147079194018056079782, 10.14613678528139183110654187343

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