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Properties

Label 2-2783-253.247-c0-0-0
Degree 22
Conductor 27832783
Sign 0.317+0.948i0.317 + 0.948i
Analytic cond. 1.388891.38889
Root an. cond. 1.178511.17851
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 − 0.225i)3-s + (−0.974 − 0.226i)4-s + (1.75 − 0.462i)5-s + (0.233 − 0.888i)9-s + (0.118 + 0.258i)12-s + (−0.410 − 0.316i)15-s + (0.897 + 0.441i)16-s + (−1.81 + 0.0519i)20-s + (−0.415 − 0.909i)23-s + (2.01 − 1.13i)25-s + (−0.503 + 0.212i)27-s + (0.949 + 1.38i)31-s + (−0.428 + 0.812i)36-s + (−0.0862 − 1.50i)37-s − 1.67i·45-s + ⋯
L(s)  = 1  + (−0.173 − 0.225i)3-s + (−0.974 − 0.226i)4-s + (1.75 − 0.462i)5-s + (0.233 − 0.888i)9-s + (0.118 + 0.258i)12-s + (−0.410 − 0.316i)15-s + (0.897 + 0.441i)16-s + (−1.81 + 0.0519i)20-s + (−0.415 − 0.909i)23-s + (2.01 − 1.13i)25-s + (−0.503 + 0.212i)27-s + (0.949 + 1.38i)31-s + (−0.428 + 0.812i)36-s + (−0.0862 − 1.50i)37-s − 1.67i·45-s + ⋯

Functional equation

Λ(s)=(2783s/2 ΓC(s) L(s)=((0.317+0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2783s/2 ΓC(s) L(s)=((0.317+0.948i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27832783    =    112⋅2311^{2} \cdot 23
Sign: 0.317+0.948i0.317 + 0.948i
Analytic conductor: 1.388891.38889
Root analytic conductor: 1.178511.17851
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2783(753,⋅)\chi_{2783} (753, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2783, ( :0), 0.317+0.948i)(2,\ 2783,\ (\ :0),\ 0.317 + 0.948i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.2774414551.277441455
L(12)L(\frac12) ≈\approx 1.2774414551.277441455
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)
bad11 1 1
23 1+(0.415+0.909i)T 1 + (0.415 + 0.909i)T
good2 1+(0.974+0.226i)T2 1 + (0.974 + 0.226i)T^{2}
3 1+(0.173+0.225i)T+(−0.254+0.967i)T2 1 + (0.173 + 0.225i)T + (-0.254 + 0.967i)T^{2}
5 1+(−1.75+0.462i)T+(0.870−0.491i)T2 1 + (-1.75 + 0.462i)T + (0.870 - 0.491i)T^{2}
7 1+(0.564−0.825i)T2 1 + (0.564 - 0.825i)T^{2}
13 1+(0.610−0.791i)T2 1 + (0.610 - 0.791i)T^{2}
17 1+(0.466−0.884i)T2 1 + (0.466 - 0.884i)T^{2}
19 1+(0.985−0.170i)T2 1 + (0.985 - 0.170i)T^{2}
29 1+(−0.985−0.170i)T2 1 + (-0.985 - 0.170i)T^{2}
31 1+(−0.949−1.38i)T+(−0.362+0.931i)T2 1 + (-0.949 - 1.38i)T + (-0.362 + 0.931i)T^{2}
37 1+(0.0862+1.50i)T+(−0.993+0.113i)T2 1 + (0.0862 + 1.50i)T + (-0.993 + 0.113i)T^{2}
41 1+(0.993+0.113i)T2 1 + (0.993 + 0.113i)T^{2}
43 1+(−0.841−0.540i)T2 1 + (-0.841 - 0.540i)T^{2}
47 1+(1.36+0.988i)T+(0.309+0.951i)T2 1 + (1.36 + 0.988i)T + (0.309 + 0.951i)T^{2}
53 1+(−0.257−1.48i)T+(−0.941+0.336i)T2 1 + (-0.257 - 1.48i)T + (-0.941 + 0.336i)T^{2}
59 1+(−0.912−0.939i)T+(−0.0285+0.999i)T2 1 + (-0.912 - 0.939i)T + (-0.0285 + 0.999i)T^{2}
61 1+(0.998+0.0570i)T2 1 + (0.998 + 0.0570i)T^{2}
67 1+(1.80+0.822i)T+(0.654+0.755i)T2 1 + (1.80 + 0.822i)T + (0.654 + 0.755i)T^{2}
71 1+(−0.612−0.561i)T+(0.0855+0.996i)T2 1 + (-0.612 - 0.561i)T + (0.0855 + 0.996i)T^{2}
73 1+(0.897+0.441i)T2 1 + (0.897 + 0.441i)T^{2}
79 1+(−0.610+0.791i)T2 1 + (-0.610 + 0.791i)T^{2}
83 1+(−0.198+0.980i)T2 1 + (-0.198 + 0.980i)T^{2}
89 1+(−0.557+1.89i)T+(−0.841−0.540i)T2 1 + (-0.557 + 1.89i)T + (-0.841 - 0.540i)T^{2}
97 1+(−0.684+0.837i)T+(−0.198−0.980i)T2 1 + (-0.684 + 0.837i)T + (-0.198 - 0.980i)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

−8.934986190208084518726379974252, −8.466872391826870273150620620393, −7.18086821212296350261120334186, −6.22758017209557366202306771889, −5.89185679406313639796174895622, −5.00256470123959033375557458038, −4.33738881330922565709499349432, −3.11793546127480029818981767865, −1.86777785633236591183643471578, −0.942106288488326087079499080449, 1.53053108737128671444430455712, 2.48762022791002308445401607171, 3.52591816037216971934412444119, 4.69916648267959934934147605080, 5.21232760367381729682983333557, 5.96309321103337113045813075060, 6.69041036806088842339703070444, 7.79810258277279810152452139122, 8.399160992048158429750473921751, 9.451653597545489715017204882832

Graph of the ZZ-function along the critical line