Properties

Label 2-756-28.19-c1-0-36
Degree $2$
Conductor $756$
Sign $0.184 + 0.982i$
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  + (0.444 − 1.34i)2-s + (−1.60 − 1.19i)4-s + (1.00 + 0.579i)5-s + (−0.250 + 2.63i)7-s + (−2.31 + 1.62i)8-s + (1.22 − 1.08i)10-s + (1.17 − 0.679i)11-s − 3.61i·13-s + (3.42 + 1.50i)14-s + (1.14 + 3.83i)16-s + (6.02 − 3.48i)17-s + (3.04 − 5.27i)19-s + (−0.917 − 2.12i)20-s + (−0.388 − 1.88i)22-s + (2.38 + 1.37i)23-s + ⋯
L(s)  = 1  + (0.314 − 0.949i)2-s + (−0.802 − 0.597i)4-s + (0.448 + 0.258i)5-s + (−0.0948 + 0.995i)7-s + (−0.819 + 0.573i)8-s + (0.386 − 0.344i)10-s + (0.355 − 0.204i)11-s − 1.00i·13-s + (0.915 + 0.403i)14-s + (0.286 + 0.957i)16-s + (1.46 − 0.844i)17-s + (0.698 − 1.20i)19-s + (−0.205 − 0.475i)20-s + (−0.0828 − 0.401i)22-s + (0.498 + 0.287i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.184 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.184 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.184 + 0.982i$
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: \((2,\ 756,\ (\ :1/2),\ 0.184 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40241 - 1.16365i\)
\(L(\frac12)\) \(\approx\) \(1.40241 - 1.16365i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.444 + 1.34i)T \)
3 \( 1 \)
7 \( 1 + (0.250 - 2.63i)T \)
good5 \( 1 + (-1.00 - 0.579i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.17 + 0.679i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.61iT - 13T^{2} \)
17 \( 1 + (-6.02 + 3.48i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.04 + 5.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.38 - 1.37i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.87T + 29T^{2} \)
31 \( 1 + (-2.09 - 3.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.83 + 3.17i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.72iT - 41T^{2} \)
43 \( 1 + 5.96iT - 43T^{2} \)
47 \( 1 + (6.26 - 10.8i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.95 - 8.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.862 - 1.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.23 + 1.28i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.37 - 3.68i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.8iT - 71T^{2} \)
73 \( 1 + (3.53 - 2.04i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.0869 + 0.0502i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.12T + 83T^{2} \)
89 \( 1 + (-2.78 - 1.61i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.23iT - 97T^{2} \)
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

−10.14466340356904923053913410476, −9.503347896701505800585603273138, −8.776584013884558338845162194126, −7.70699140380570247845470452227, −6.30628443751869821224494609936, −5.51259863602002685908438496394, −4.78070494486077456407517283551, −3.12204244102426866457381092730, −2.73073843907995619600607565490, −1.05648691155342622498166105173, 1.33715953648965224949913939281, 3.43204186347910154502678816560, 4.20380085808199901931931157167, 5.27412610879449936096507566260, 6.17895334910250782675171412534, 6.99453805299817240903491814370, 7.82648808191720719152742815977, 8.621366392538333134086724432051, 9.827079382170149978866378483550, 10.02933236176758226224038898762

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