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Properties

Label 2-2280-2280.149-c0-0-2
Degree 22
Conductor 22802280
Sign −0.356−0.934i-0.356 - 0.934i
Analytic cond. 1.137861.13786
Root an. cond. 1.066701.06670
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  + (0.866 + 0.5i)2-s + (0.342 + 0.939i)3-s + (0.499 + 0.866i)4-s + (0.984 + 0.173i)5-s + (−0.173 + 0.984i)6-s + 0.999i·8-s + (−0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (−0.642 + 0.766i)12-s + (0.173 + 0.984i)15-s + (−0.5 + 0.866i)16-s + (−0.984 − 0.826i)17-s + (−0.984 + 0.173i)18-s + (0.939 − 0.342i)19-s + (0.342 + 0.939i)20-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.342 + 0.939i)3-s + (0.499 + 0.866i)4-s + (0.984 + 0.173i)5-s + (−0.173 + 0.984i)6-s + 0.999i·8-s + (−0.766 + 0.642i)9-s + (0.766 + 0.642i)10-s + (−0.642 + 0.766i)12-s + (0.173 + 0.984i)15-s + (−0.5 + 0.866i)16-s + (−0.984 − 0.826i)17-s + (−0.984 + 0.173i)18-s + (0.939 − 0.342i)19-s + (0.342 + 0.939i)20-s + ⋯

Functional equation

Λ(s)=(2280s/2 ΓC(s) L(s)=((−0.356−0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2280s/2 ΓC(s) L(s)=((−0.356−0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 22802280    =    23⋅3⋅5⋅192^{3} \cdot 3 \cdot 5 \cdot 19
Sign: −0.356−0.934i-0.356 - 0.934i
Analytic conductor: 1.137861.13786
Root analytic conductor: 1.066701.06670
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2280(149,⋅)\chi_{2280} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2280, ( :0), −0.356−0.934i)(2,\ 2280,\ (\ :0),\ -0.356 - 0.934i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.4615989252.461598925
L(12)L(\frac12) ≈\approx 2.4615989252.461598925
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+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
3 1+(−0.342−0.939i)T 1 + (-0.342 - 0.939i)T
5 1+(−0.984−0.173i)T 1 + (-0.984 - 0.173i)T
19 1+(−0.939+0.342i)T 1 + (-0.939 + 0.342i)T
good7 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+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
17 1+(0.984+0.826i)T+(0.173+0.984i)T2 1 + (0.984 + 0.826i)T + (0.173 + 0.984i)T^{2}
23 1+(0.300+1.70i)T+(−0.939+0.342i)T2 1 + (0.300 + 1.70i)T + (-0.939 + 0.342i)T^{2}
29 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
31 1+(0.939+1.62i)T+(−0.5+0.866i)T2 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2}
37 1+T2 1 + T^{2}
41 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
43 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
47 1+(1.50−1.26i)T+(0.173−0.984i)T2 1 + (1.50 - 1.26i)T + (0.173 - 0.984i)T^{2}
53 1+(−1.50+0.266i)T+(0.939−0.342i)T2 1 + (-1.50 + 0.266i)T + (0.939 - 0.342i)T^{2}
59 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
61 1+(−1.70+0.300i)T+(0.939−0.342i)T2 1 + (-1.70 + 0.300i)T + (0.939 - 0.342i)T^{2}
67 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
71 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
73 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
79 1+(−0.939+0.342i)T+(0.766−0.642i)T2 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}
83 1+(1.62−0.939i)T+(0.5−0.866i)T2 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2}
89 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
97 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)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

−9.399313899642174710785059268439, −8.728064614999930242179623753277, −7.87490611006388416721084217075, −6.91651388262979412051576383986, −6.18609382587339390639499307910, −5.36045570161972751336172682130, −4.74112316356365559276468352395, −3.92988207697281152872630394166, −2.79730037765058146652185456382, −2.28795269865614845689454342154, 1.44231220797025604146390422432, 1.96651756095480005640646469026, 3.07952420133968703780168928538, 3.86689627186954965792043004105, 5.33187860016042714854797281223, 5.57684232631362716036573898427, 6.67008742371356134207488058035, 7.03984949762457948554558267362, 8.228632700090879533218347987776, 9.057133642723107935778576871246

Graph of the ZZ-function along the critical line