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Properties

Label 2-2312-136.43-c0-0-5
Degree 22
Conductor 23122312
Sign 0.0340−0.999i0.0340 - 0.999i
Analytic cond. 1.153831.15383
Root an. cond. 1.074161.07416
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  + (0.707 − 0.707i)2-s + (−1.30 − 0.541i)3-s − 1.00i·4-s + (−1.30 + 0.541i)6-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−1.30 + 0.541i)11-s + (−0.541 + 1.30i)12-s − 1.00·16-s + 1.00·18-s + (−1.41 + 1.41i)19-s + (−0.541 + 1.30i)22-s + (0.541 + 1.30i)24-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)32-s + 2·33-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−1.30 − 0.541i)3-s − 1.00i·4-s + (−1.30 + 0.541i)6-s + (−0.707 − 0.707i)8-s + (0.707 + 0.707i)9-s + (−1.30 + 0.541i)11-s + (−0.541 + 1.30i)12-s − 1.00·16-s + 1.00·18-s + (−1.41 + 1.41i)19-s + (−0.541 + 1.30i)22-s + (0.541 + 1.30i)24-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.707i)32-s + 2·33-s + ⋯

Functional equation

Λ(s)=(2312s/2 ΓC(s) L(s)=((0.0340−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0340 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2312s/2 ΓC(s) L(s)=((0.0340−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0340 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23122312    =    23⋅1722^{3} \cdot 17^{2}
Sign: 0.0340−0.999i0.0340 - 0.999i
Analytic conductor: 1.153831.15383
Root analytic conductor: 1.074161.07416
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2312(179,⋅)\chi_{2312} (179, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2312, ( :0), 0.0340−0.999i)(2,\ 2312,\ (\ :0),\ 0.0340 - 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.019178717150.01917871715
L(12)L(\frac12) ≈\approx 0.019178717150.01917871715
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+(−0.707+0.707i)T 1 + (-0.707 + 0.707i)T
17 1 1
good3 1+(1.30+0.541i)T+(0.707+0.707i)T2 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2}
5 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
7 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
11 1+(1.30−0.541i)T+(0.707−0.707i)T2 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2}
13 1+T2 1 + T^{2}
19 1+(1.41−1.41i)T−iT2 1 + (1.41 - 1.41i)T - iT^{2}
23 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
29 1+(0.707+0.707i)T2 1 + (0.707 + 0.707i)T^{2}
31 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
37 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
41 1+(0.541+1.30i)T+(−0.707+0.707i)T2 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2}
43 1+iT2 1 + iT^{2}
47 1+T2 1 + T^{2}
53 1+iT2 1 + iT^{2}
59 1+iT2 1 + iT^{2}
61 1+(0.707−0.707i)T2 1 + (0.707 - 0.707i)T^{2}
67 1+T2 1 + T^{2}
71 1+(−0.707−0.707i)T2 1 + (-0.707 - 0.707i)T^{2}
73 1+(0.541−1.30i)T+(−0.707−0.707i)T2 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2}
79 1+(−0.707+0.707i)T2 1 + (-0.707 + 0.707i)T^{2}
83 1−iT2 1 - iT^{2}
89 1−T2 1 - T^{2}
97 1+(0.541−1.30i)T+(−0.707−0.707i)T2 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)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

−8.582051775693450735431189984296, −7.66092736527158021305948694791, −6.73672510411999652078575039144, −6.02392477784236726347682157182, −5.48469666911059350348035552594, −4.69812673536794768578245813718, −3.84233502256158039328740147862, −2.49208525649374101695453126519, −1.62388067802965991236320491340, −0.01135336957170926200202113786, 2.43862206839291801959350529122, 3.50559486597387139909327336238, 4.67265521073079091434367388678, 4.95497736745412027062781900266, 5.85461392056205052551439665256, 6.35930399238639030035931046177, 7.23053804079752615670303589175, 8.112694572185609098480151016123, 8.823031235493531295420680526626, 9.883702062746596924560779573303

Graph of the ZZ-function along the critical line