↑

Properties

Label 2-756-28.19-c1-0-23
Degree $2$
Conductor $756$
Sign $0.972 + 0.232i$
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  + (0.370 − 1.36i)2-s + (−1.72 − 1.01i)4-s + (3.53 + 2.04i)5-s + (2.14 + 1.54i)7-s + (−2.01 + 1.98i)8-s + (4.09 − 4.07i)10-s + (−2.02 + 1.16i)11-s + 0.570i·13-s + (2.90 − 2.35i)14-s + (1.95 + 3.48i)16-s + (1.25 − 0.727i)17-s + (−3.88 + 6.72i)19-s + (−4.04 − 7.10i)20-s + (0.844 + 3.18i)22-s + (2.14 + 1.23i)23-s + ⋯
L(s)  = 1  + (0.261 − 0.965i)2-s + (−0.863 − 0.505i)4-s + (1.58 + 0.913i)5-s + (0.810 + 0.585i)7-s + (−0.713 + 0.700i)8-s + (1.29 − 1.28i)10-s + (−0.609 + 0.351i)11-s + 0.158i·13-s + (0.777 − 0.629i)14-s + (0.489 + 0.871i)16-s + (0.305 − 0.176i)17-s + (−0.890 + 1.54i)19-s + (−0.904 − 1.58i)20-s + (0.180 + 0.679i)22-s + (0.447 + 0.258i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.972 + 0.232i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.972 + 0.232i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10089 - 0.247512i\)
\(L(\frac12)\) \(\approx\) \(2.10089 - 0.247512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.370 + 1.36i)T \)
3 \( 1 \)
7 \( 1 + (-2.14 - 1.54i)T \)
good5 \( 1 + (-3.53 - 2.04i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.02 - 1.16i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.570iT - 13T^{2} \)
17 \( 1 + (-1.25 + 0.727i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.88 - 6.72i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.14 - 1.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 4.52T + 29T^{2} \)
31 \( 1 + (3.30 + 5.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.68 - 4.64i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 + 5.76iT - 43T^{2} \)
47 \( 1 + (-4.39 + 7.62i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.16 + 7.21i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.17 + 3.77i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.64 + 3.83i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.12 + 2.38i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.60iT - 71T^{2} \)
73 \( 1 + (-1.90 + 1.09i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.75 - 5.05i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.1T + 83T^{2} \)
89 \( 1 + (3.41 + 1.96i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.3iT - 97T^{2} \)
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

−10.46564285126221484842336654593, −9.728642387081076287909469049737, −8.929128671028531565524959445443, −7.898053479914703417521305168724, −6.50928323152768845414166462295, −5.62908554171482120763034902226, −5.04382699843433025528897360561, −3.55779716212664322200685465701, −2.28839902222921047323587815743, −1.83992548057989806157895178704, 1.09590327112594398364049850480, 2.73413934586451816320087184185, 4.55518454107288339154862362287, 4.98294746346133742944580439289, 5.89491621189229051792796538848, 6.72198280881000077625761792885, 7.82465714862405451936120952980, 8.685776037850318303905428820600, 9.207809589937964600809635938400, 10.24513181093775520816928143937

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