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Properties

Label 2-756-12.11-c1-0-31
Degree $2$
Conductor $756$
Sign $-0.668 + 0.743i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.575 − 1.29i)2-s + (−1.33 + 1.48i)4-s − 0.311i·5-s − i·7-s + (2.69 + 0.870i)8-s + (−0.402 + 0.179i)10-s + 3.59·11-s − 6.44·13-s + (−1.29 + 0.575i)14-s + (−0.425 − 3.97i)16-s − 2.96i·17-s − 0.684i·19-s + (0.463 + 0.416i)20-s + (−2.06 − 4.64i)22-s + 7.58·23-s + ⋯
L(s)  = 1  + (−0.407 − 0.913i)2-s + (−0.668 + 0.743i)4-s − 0.139i·5-s − 0.377i·7-s + (0.951 + 0.307i)8-s + (−0.127 + 0.0566i)10-s + 1.08·11-s − 1.78·13-s + (−0.345 + 0.153i)14-s + (−0.106 − 0.994i)16-s − 0.718i·17-s − 0.157i·19-s + (0.103 + 0.0930i)20-s + (−0.441 − 0.989i)22-s + 1.58·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-0.668 + 0.743i$
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: \((2,\ 756,\ (\ :1/2),\ -0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.397325 - 0.891247i\)
\(L(\frac12)\) \(\approx\) \(0.397325 - 0.891247i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.575 + 1.29i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 0.311iT - 5T^{2} \)
11 \( 1 - 3.59T + 11T^{2} \)
13 \( 1 + 6.44T + 13T^{2} \)
17 \( 1 + 2.96iT - 17T^{2} \)
19 \( 1 + 0.684iT - 19T^{2} \)
23 \( 1 - 7.58T + 23T^{2} \)
29 \( 1 + 9.26iT - 29T^{2} \)
31 \( 1 + 1.91iT - 31T^{2} \)
37 \( 1 + 8.40T + 37T^{2} \)
41 \( 1 + 7.50iT - 41T^{2} \)
43 \( 1 + 9.30iT - 43T^{2} \)
47 \( 1 + 9.82T + 47T^{2} \)
53 \( 1 - 5.43iT - 53T^{2} \)
59 \( 1 - 2.29T + 59T^{2} \)
61 \( 1 - 0.486T + 61T^{2} \)
67 \( 1 + 1.09iT - 67T^{2} \)
71 \( 1 - 1.76T + 71T^{2} \)
73 \( 1 - 3.32T + 73T^{2} \)
79 \( 1 + 1.39iT - 79T^{2} \)
83 \( 1 - 0.283T + 83T^{2} \)
89 \( 1 + 8.91iT - 89T^{2} \)
97 \( 1 - 0.621T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.967245833283048895079922833238, −9.336651185424021729414595770898, −8.635483412345095063825273654887, −7.40278035768477964876842413809, −6.91344610489647471275708081122, −5.16338146648180794258916666138, −4.44257085734102208000461870194, −3.27148802939068899768744869005, −2.15327812287703957054281207279, −0.61399094818199120745878395068, 1.47117084285925824967389574865, 3.18904108421566981822308131918, 4.70402005854135994144043937074, 5.27065220205066789388519910566, 6.68120490880962050420800938818, 6.92231974760026463963793147524, 8.100701460886292839174726796741, 8.950002936209899753064991979717, 9.551821257672279584796008752011, 10.42796350879099536365361885233

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