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Properties

Label 2-3380-3380.383-c0-0-0
Degree 22
Conductor 33803380
Sign 0.818−0.573i0.818 - 0.573i
Analytic cond. 1.686831.68683
Root an. cond. 1.298781.29878
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  + (0.316 + 0.948i)2-s + (−0.799 + 0.600i)4-s + (0.428 − 0.903i)5-s + (−0.822 − 0.568i)8-s + (−0.979 − 0.200i)9-s + (0.992 + 0.120i)10-s + (0.996 − 0.0804i)13-s + (0.278 − 0.960i)16-s + (1.28 + 1.50i)17-s + (−0.120 − 0.992i)18-s + (0.200 + 0.979i)20-s + (−0.632 − 0.774i)25-s + (0.391 + 0.919i)26-s + (−0.625 − 1.87i)29-s + (0.999 − 0.0402i)32-s + ⋯
L(s)  = 1  + (0.316 + 0.948i)2-s + (−0.799 + 0.600i)4-s + (0.428 − 0.903i)5-s + (−0.822 − 0.568i)8-s + (−0.979 − 0.200i)9-s + (0.992 + 0.120i)10-s + (0.996 − 0.0804i)13-s + (0.278 − 0.960i)16-s + (1.28 + 1.50i)17-s + (−0.120 − 0.992i)18-s + (0.200 + 0.979i)20-s + (−0.632 − 0.774i)25-s + (0.391 + 0.919i)26-s + (−0.625 − 1.87i)29-s + (0.999 − 0.0402i)32-s + ⋯

Functional equation

Λ(s)=(3380s/2 ΓC(s) L(s)=((0.818−0.573i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3380s/2 ΓC(s) L(s)=((0.818−0.573i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.818 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33803380    =    22⋅5⋅1322^{2} \cdot 5 \cdot 13^{2}
Sign: 0.818−0.573i0.818 - 0.573i
Analytic conductor: 1.686831.68683
Root analytic conductor: 1.298781.29878
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3380(383,⋅)\chi_{3380} (383, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3380, ( :0), 0.818−0.573i)(2,\ 3380,\ (\ :0),\ 0.818 - 0.573i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4203014101.420301410
L(12)L(\frac12) ≈\approx 1.4203014101.420301410
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+(−0.316−0.948i)T 1 + (-0.316 - 0.948i)T
5 1+(−0.428+0.903i)T 1 + (-0.428 + 0.903i)T
13 1+(−0.996+0.0804i)T 1 + (-0.996 + 0.0804i)T
good3 1+(0.979+0.200i)T2 1 + (0.979 + 0.200i)T^{2}
7 1+(−0.987−0.160i)T2 1 + (-0.987 - 0.160i)T^{2}
11 1+(0.391+0.919i)T2 1 + (0.391 + 0.919i)T^{2}
17 1+(−1.28−1.50i)T+(−0.160+0.987i)T2 1 + (-1.28 - 1.50i)T + (-0.160 + 0.987i)T^{2}
19 1+(0.866+0.5i)T2 1 + (0.866 + 0.5i)T^{2}
23 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
29 1+(0.625+1.87i)T+(−0.799+0.600i)T2 1 + (0.625 + 1.87i)T + (-0.799 + 0.600i)T^{2}
31 1+(0.822+0.568i)T2 1 + (0.822 + 0.568i)T^{2}
37 1+(−0.0580+1.44i)T+(−0.996−0.0804i)T2 1 + (-0.0580 + 1.44i)T + (-0.996 - 0.0804i)T^{2}
41 1+(−1.78−0.179i)T+(0.979+0.200i)T2 1 + (-1.78 - 0.179i)T + (0.979 + 0.200i)T^{2}
43 1+(−0.0804−0.996i)T2 1 + (-0.0804 - 0.996i)T^{2}
47 1+(0.970+0.239i)T2 1 + (0.970 + 0.239i)T^{2}
53 1+(−1.49+0.274i)T+(0.935−0.354i)T2 1 + (-1.49 + 0.274i)T + (0.935 - 0.354i)T^{2}
59 1+(−0.534−0.845i)T2 1 + (-0.534 - 0.845i)T^{2}
61 1+(0.664−1.39i)T+(−0.632−0.774i)T2 1 + (0.664 - 1.39i)T + (-0.632 - 0.774i)T^{2}
67 1+(0.278+0.960i)T2 1 + (0.278 + 0.960i)T^{2}
71 1+(0.316+0.948i)T2 1 + (0.316 + 0.948i)T^{2}
73 1+(−0.663−0.748i)T+(−0.120+0.992i)T2 1 + (-0.663 - 0.748i)T + (-0.120 + 0.992i)T^{2}
79 1+(−0.970−0.239i)T2 1 + (-0.970 - 0.239i)T^{2}
83 1+(0.748−0.663i)T2 1 + (0.748 - 0.663i)T^{2}
89 1+(−0.267−0.999i)T+(−0.866+0.5i)T2 1 + (-0.267 - 0.999i)T + (-0.866 + 0.5i)T^{2}
97 1+(−0.670+0.643i)T+(0.0402−0.999i)T2 1 + (-0.670 + 0.643i)T + (0.0402 - 0.999i)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

−8.717550503802671265876487034549, −8.105890817115715791891659481479, −7.56790742653017248779721540850, −6.26372291721405183492315688677, −5.73277662279346476540879444279, −5.55815531920154732676483867057, −4.13782111858846726055455030178, −3.80496597957116759505926663948, −2.47169665925952121950307818258, −0.943251840937850096302143508298, 1.17244578880822093606922565113, 2.41410101397926886016974387259, 3.11273183798426328358523820920, 3.66540100225384444799281919250, 4.99604580051880279808347901541, 5.57423171102884303452538647193, 6.23806020771007030577987685460, 7.21324585374728842745521430017, 8.085422669194973100493695881582, 9.053740472708010566802146051375

Graph of the ZZ-function along the critical line