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Properties

Label 2-756-21.11-c0-0-0
Degree 22
Conductor 756756
Sign 0.895−0.444i0.895 - 0.444i
Analytic cond. 0.3772930.377293
Root an. cond. 0.6142410.614241
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)7-s + 2·13-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)31-s + (−1 − 1.73i)37-s − 43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (0.5 − 0.866i)73-s + (−1 − 1.73i)79-s + (−1 + 1.73i)91-s − 97-s + (−1 − 1.73i)103-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + 2·13-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)31-s + (−1 − 1.73i)37-s − 43-s + (−0.499 − 0.866i)49-s + (0.5 + 0.866i)61-s + (−1 + 1.73i)67-s + (0.5 − 0.866i)73-s + (−1 − 1.73i)79-s + (−1 + 1.73i)91-s − 97-s + (−1 − 1.73i)103-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 756756    =    22⋅33⋅72^{2} \cdot 3^{3} \cdot 7
Sign: 0.895−0.444i0.895 - 0.444i
Analytic conductor: 0.3772930.377293
Root analytic conductor: 0.6142410.614241
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ756(53,⋅)\chi_{756} (53, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 756, ( :0), 0.895−0.444i)(2,\ 756,\ (\ :0),\ 0.895 - 0.444i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.97250353910.9725035391
L(12)L(\frac12) ≈\approx 0.97250353910.9725035391
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
7 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
13 1−2T+T2 1 - 2T + T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
37 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+T+T2 1 + T + T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−10.62932871528426230517026056625, −9.687532573179586339968001977749, −8.870239416831465612952893521537, −8.232794775295628069526212849968, −7.10171719857339533757056547881, −5.96747182206304400887987792285, −5.61043948651723752723535650071, −4.00545553845051295599186923696, −3.18449586044372927020330116682, −1.68767077795690034784893235093, 1.26918191813805124151530000226, 3.10468264816957414902063507855, 3.92259897083775955625535401179, 5.04877944921169936566024468690, 6.34137702286563838020390891255, 6.78871695268480433648327619841, 8.034532254963809259935478754537, 8.699164035819916900282065218251, 9.742232412987617180579576569954, 10.50220222348165524604662537221

Graph of the ZZ-function along the critical line