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Properties

Label 2-3040-760.339-c0-0-1
Degree 22
Conductor 30403040
Sign 0.513+0.858i0.513 + 0.858i
Analytic cond. 1.517151.51715
Root an. cond. 1.231721.23172
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  + (0.173 − 0.984i)5-s + (0.173 + 0.300i)7-s + (0.766 − 0.642i)9-s + (0.766 − 1.32i)11-s + (0.939 + 0.342i)13-s + (−0.766 − 0.642i)19-s + (0.326 + 1.85i)23-s + (−0.939 − 0.342i)25-s + (0.326 − 0.118i)35-s − 1.87·37-s + (−1.43 + 0.524i)41-s + (−0.5 − 0.866i)45-s + (0.766 − 0.642i)47-s + (0.439 − 0.761i)49-s + (0.0603 + 0.342i)53-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)5-s + (0.173 + 0.300i)7-s + (0.766 − 0.642i)9-s + (0.766 − 1.32i)11-s + (0.939 + 0.342i)13-s + (−0.766 − 0.642i)19-s + (0.326 + 1.85i)23-s + (−0.939 − 0.342i)25-s + (0.326 − 0.118i)35-s − 1.87·37-s + (−1.43 + 0.524i)41-s + (−0.5 − 0.866i)45-s + (0.766 − 0.642i)47-s + (0.439 − 0.761i)49-s + (0.0603 + 0.342i)53-s + ⋯

Functional equation

Λ(s)=(3040s/2 ΓC(s) L(s)=((0.513+0.858i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3040s/2 ΓC(s) L(s)=((0.513+0.858i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 30403040    =    25⋅5⋅192^{5} \cdot 5 \cdot 19
Sign: 0.513+0.858i0.513 + 0.858i
Analytic conductor: 1.517151.51715
Root analytic conductor: 1.231721.23172
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3040(719,⋅)\chi_{3040} (719, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3040, ( :0), 0.513+0.858i)(2,\ 3040,\ (\ :0),\ 0.513 + 0.858i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4669229561.466922956
L(12)L(\frac12) ≈\approx 1.4669229561.466922956
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
5 1+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
19 1+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
good3 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
7 1+(−0.173−0.300i)T+(−0.5+0.866i)T2 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2}
11 1+(−0.766+1.32i)T+(−0.5−0.866i)T2 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2}
13 1+(−0.939−0.342i)T+(0.766+0.642i)T2 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}
17 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
23 1+(−0.326−1.85i)T+(−0.939+0.342i)T2 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2}
29 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
31 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
37 1+1.87T+T2 1 + 1.87T + T^{2}
41 1+(1.43−0.524i)T+(0.766−0.642i)T2 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2}
43 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
47 1+(−0.766+0.642i)T+(0.173−0.984i)T2 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2}
53 1+(−0.0603−0.342i)T+(−0.939+0.342i)T2 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2}
59 1+(−0.766−0.642i)T+(0.173+0.984i)T2 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2}
61 1+(0.939−0.342i)T2 1 + (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+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
83 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
89 1+(−1.76−0.642i)T+(0.766+0.642i)T2 1 + (-1.76 - 0.642i)T + (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

−8.831204430282686151048658629073, −8.356167425987301212531909120440, −7.18422841045137071596650748255, −6.45647445316805296303329126534, −5.71473363315743450351330412483, −4.99811793399565669109892879695, −3.89127048210134237476464856685, −3.48350150715923967869680499527, −1.81483610072847258740953384780, −1.03188143779678336927879676959, 1.58810479876707496338393692004, 2.30955032332008877390669983433, 3.59697862205992056583519258690, 4.25039907990385125908785011267, 5.08545707655593841234401234439, 6.26235021484750890794680281067, 6.79827287123337314864700455823, 7.35525529860549122109266265075, 8.230914597813344408081200321619, 8.979677881980374828421181767864

Graph of the ZZ-function along the critical line