↑

Properties

Label 2-756-12.11-c1-0-21
Degree $2$
Conductor $756$
Sign $0.226 - 0.974i$
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  + (1.40 + 0.161i)2-s + (1.94 + 0.452i)4-s + 4.14i·5-s − i·7-s + (2.66 + 0.950i)8-s + (−0.668 + 5.83i)10-s − 3.98·11-s + 5.19·13-s + (0.161 − 1.40i)14-s + (3.58 + 1.76i)16-s − 0.533i·17-s − 1.00i·19-s + (−1.87 + 8.08i)20-s + (−5.59 − 0.641i)22-s − 2.52·23-s + ⋯
L(s)  = 1  + (0.993 + 0.113i)2-s + (0.974 + 0.226i)4-s + 1.85i·5-s − 0.377i·7-s + (0.941 + 0.336i)8-s + (−0.211 + 1.84i)10-s − 1.20·11-s + 1.44·13-s + (0.0430 − 0.375i)14-s + (0.897 + 0.441i)16-s − 0.129i·17-s − 0.231i·19-s + (−0.420 + 1.80i)20-s + (−1.19 − 0.136i)22-s − 0.526·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.226 - 0.974i$
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: \((2,\ 756,\ (\ :1/2),\ 0.226 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21221 + 1.75686i\)
\(L(\frac12)\) \(\approx\) \(2.21221 + 1.75686i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.161i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 4.14iT - 5T^{2} \)
11 \( 1 + 3.98T + 11T^{2} \)
13 \( 1 - 5.19T + 13T^{2} \)
17 \( 1 + 0.533iT - 17T^{2} \)
19 \( 1 + 1.00iT - 19T^{2} \)
23 \( 1 + 2.52T + 23T^{2} \)
29 \( 1 - 8.68iT - 29T^{2} \)
31 \( 1 - 5.49iT - 31T^{2} \)
37 \( 1 - 2.40T + 37T^{2} \)
41 \( 1 + 8.54iT - 41T^{2} \)
43 \( 1 + 10.7iT - 43T^{2} \)
47 \( 1 - 9.11T + 47T^{2} \)
53 \( 1 + 7.18iT - 53T^{2} \)
59 \( 1 - 6.12T + 59T^{2} \)
61 \( 1 - 1.85T + 61T^{2} \)
67 \( 1 + 11.3iT - 67T^{2} \)
71 \( 1 + 8.36T + 71T^{2} \)
73 \( 1 + 1.90T + 73T^{2} \)
79 \( 1 + 1.66iT - 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + 3.43iT - 89T^{2} \)
97 \( 1 - 0.254T + 97T^{2} \)
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

−10.61671781681948360252229706185, −10.32832582913579514033551521616, −8.593588505601114550503609658714, −7.48197464532544933687139197391, −6.98675989560446912343929456767, −6.13349075212969862093540150529, −5.27570545610624150931558573558, −3.79691107864535626517961746329, −3.20689295518683957658533669052, −2.14055638754275085828986658447, 1.13001388665605354675743689912, 2.45085259487442597567122001316, 3.98230661213202043028800597607, 4.64338736997297309724830534173, 5.72277236783257634728961287904, 6.03261845983798897092244745939, 7.84967119876984243397833896274, 8.207066168556532231298421358660, 9.330934082567673991806599675396, 10.21926289195602899107221512490

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