Properties

Label 2-3680-115.84-c0-0-1
Degree 22
Conductor 36803680
Sign 0.8080.588i0.808 - 0.588i
Analytic cond. 1.836551.83655
Root an. cond. 1.355191.35519
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  + (1.09 − 0.497i)3-s + (−0.755 + 0.654i)5-s + (1.64 + 1.05i)7-s + (0.285 − 0.329i)9-s + (−0.497 + 1.09i)15-s + (2.31 + 0.333i)21-s + (0.599 + 0.800i)23-s + (0.142 − 0.989i)25-s + (−0.190 + 0.647i)27-s + (−1.25 + 0.368i)29-s + (−1.93 + 0.278i)35-s + (−0.708 − 0.817i)41-s + (−0.729 − 1.59i)43-s + 0.436i·45-s + 0.142i·47-s + ⋯
L(s)  = 1  + (1.09 − 0.497i)3-s + (−0.755 + 0.654i)5-s + (1.64 + 1.05i)7-s + (0.285 − 0.329i)9-s + (−0.497 + 1.09i)15-s + (2.31 + 0.333i)21-s + (0.599 + 0.800i)23-s + (0.142 − 0.989i)25-s + (−0.190 + 0.647i)27-s + (−1.25 + 0.368i)29-s + (−1.93 + 0.278i)35-s + (−0.708 − 0.817i)41-s + (−0.729 − 1.59i)43-s + 0.436i·45-s + 0.142i·47-s + ⋯

Functional equation

Λ(s)=(3680s/2ΓC(s)L(s)=((0.8080.588i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3680s/2ΓC(s)L(s)=((0.8080.588i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.588i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36803680    =    255232^{5} \cdot 5 \cdot 23
Sign: 0.8080.588i0.808 - 0.588i
Analytic conductor: 1.836551.83655
Root analytic conductor: 1.355191.35519
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3680(3649,)\chi_{3680} (3649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3680, ( :0), 0.8080.588i)(2,\ 3680,\ (\ :0),\ 0.808 - 0.588i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.9061156321.906115632
L(12)L(\frac12) \approx 1.9061156321.906115632
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1+(0.7550.654i)T 1 + (0.755 - 0.654i)T
23 1+(0.5990.800i)T 1 + (-0.599 - 0.800i)T
good3 1+(1.09+0.497i)T+(0.6540.755i)T2 1 + (-1.09 + 0.497i)T + (0.654 - 0.755i)T^{2}
7 1+(1.641.05i)T+(0.415+0.909i)T2 1 + (-1.64 - 1.05i)T + (0.415 + 0.909i)T^{2}
11 1+(0.9590.281i)T2 1 + (0.959 - 0.281i)T^{2}
13 1+(0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
17 1+(0.8410.540i)T2 1 + (0.841 - 0.540i)T^{2}
19 1+(0.8410.540i)T2 1 + (-0.841 - 0.540i)T^{2}
29 1+(1.250.368i)T+(0.8410.540i)T2 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2}
31 1+(0.6540.755i)T2 1 + (-0.654 - 0.755i)T^{2}
37 1+(0.1420.989i)T2 1 + (-0.142 - 0.989i)T^{2}
41 1+(0.708+0.817i)T+(0.142+0.989i)T2 1 + (0.708 + 0.817i)T + (-0.142 + 0.989i)T^{2}
43 1+(0.729+1.59i)T+(0.654+0.755i)T2 1 + (0.729 + 1.59i)T + (-0.654 + 0.755i)T^{2}
47 10.142iTT2 1 - 0.142iT - T^{2}
53 1+(0.415+0.909i)T2 1 + (0.415 + 0.909i)T^{2}
59 1+(0.4150.909i)T2 1 + (0.415 - 0.909i)T^{2}
61 1+(1.740.797i)T+(0.654+0.755i)T2 1 + (-1.74 - 0.797i)T + (0.654 + 0.755i)T^{2}
67 1+(0.0994+0.691i)T+(0.9590.281i)T2 1 + (-0.0994 + 0.691i)T + (-0.959 - 0.281i)T^{2}
71 1+(0.9590.281i)T2 1 + (-0.959 - 0.281i)T^{2}
73 1+(0.8410.540i)T2 1 + (-0.841 - 0.540i)T^{2}
79 1+(0.415+0.909i)T2 1 + (-0.415 + 0.909i)T^{2}
83 1+(1.221.41i)T+(0.1420.989i)T2 1 + (1.22 - 1.41i)T + (-0.142 - 0.989i)T^{2}
89 1+(1.80+0.822i)T+(0.6540.755i)T2 1 + (-1.80 + 0.822i)T + (0.654 - 0.755i)T^{2}
97 1+(0.142+0.989i)T2 1 + (-0.142 + 0.989i)T^{2}
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   L(s)=p j=12(1αj,pps)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

−8.650954471839809099903837315370, −8.086852005555389594001419695601, −7.44592784427851121662415834535, −6.98048495709471603173783506390, −5.63567856610402714854916817719, −5.09143248982754611236212811115, −3.97195211751783610767450591377, −3.17398191815913466890304235639, −2.27681551071833359156465535114, −1.66236203291104878264864555739, 1.05224602767372644078808664114, 2.12630305605077542984929108322, 3.37463734937184264094208059405, 4.02743116300192030575928637650, 4.65590458789864105735356932556, 5.21878626311066504081244890958, 6.62661397917501052559172023170, 7.62194398943483377381729460108, 7.947537552807281787971267507959, 8.529505867260315527596468832973

Graph of the ZZ-function along the critical line