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Properties

Label 2-1872-156.119-c0-0-1
Degree 22
Conductor 18721872
Sign 0.556+0.831i0.556 + 0.831i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
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)5-s + (0.5 + 0.866i)13-s + (0.965 − 1.67i)17-s + (−0.448 + 0.258i)29-s + (0.5 − 1.86i)37-s + (0.448 − 1.67i)41-s + (0.866 − 0.5i)49-s + 0.517i·53-s + (−0.5 + 0.866i)61-s + (0.258 − 0.965i)65-s + (−0.366 − 0.366i)73-s + (−1.86 + 0.5i)85-s + (−1.93 − 0.517i)89-s + (0.366 + 1.36i)97-s + (0.965 + 1.67i)101-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s + (0.5 + 0.866i)13-s + (0.965 − 1.67i)17-s + (−0.448 + 0.258i)29-s + (0.5 − 1.86i)37-s + (0.448 − 1.67i)41-s + (0.866 − 0.5i)49-s + 0.517i·53-s + (−0.5 + 0.866i)61-s + (0.258 − 0.965i)65-s + (−0.366 − 0.366i)73-s + (−1.86 + 0.5i)85-s + (−1.93 − 0.517i)89-s + (0.366 + 1.36i)97-s + (0.965 + 1.67i)101-s + ⋯

Functional equation

Λ(s)=(1872s/2 ΓC(s) L(s)=((0.556+0.831i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1872s/2 ΓC(s) L(s)=((0.556+0.831i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18721872    =    24⋅32⋅132^{4} \cdot 3^{2} \cdot 13
Sign: 0.556+0.831i0.556 + 0.831i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1872(431,⋅)\chi_{1872} (431, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1872, ( :0), 0.556+0.831i)(2,\ 1872,\ (\ :0),\ 0.556 + 0.831i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0032779671.003277967
L(12)L(\frac12) ≈\approx 1.0032779671.003277967
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
3 1 1
13 1+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
good5 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
7 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
11 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
17 1+(−0.965+1.67i)T+(−0.5−0.866i)T2 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2}
19 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(0.448−0.258i)T+(0.5−0.866i)T2 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2}
31 1+iT2 1 + iT^{2}
37 1+(−0.5+1.86i)T+(−0.866−0.5i)T2 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2}
41 1+(−0.448+1.67i)T+(−0.866−0.5i)T2 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2}
43 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
47 1+iT2 1 + iT^{2}
53 1−0.517iT−T2 1 - 0.517iT - T^{2}
59 1+(0.866−0.5i)T2 1 + (0.866 - 0.5i)T^{2}
61 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
67 1+(−0.866−0.5i)T2 1 + (-0.866 - 0.5i)T^{2}
71 1+(−0.866+0.5i)T2 1 + (-0.866 + 0.5i)T^{2}
73 1+(0.366+0.366i)T+iT2 1 + (0.366 + 0.366i)T + iT^{2}
79 1−T2 1 - T^{2}
83 1−iT2 1 - iT^{2}
89 1+(1.93+0.517i)T+(0.866+0.5i)T2 1 + (1.93 + 0.517i)T + (0.866 + 0.5i)T^{2}
97 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)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

−9.061061646021698896524688568300, −8.746379215077930883877215070430, −7.51605458738415921466398153742, −7.28312919271142887458709880223, −5.99912894979101797793164851256, −5.20287581705692144380683655781, −4.31816469667403411058172394207, −3.58655559361355808467770415431, −2.31948397193740713601295863855, −0.843146133039978458504499634867, 1.40845621313387123612021937459, 2.96798788247115091037400467971, 3.56116110526454750872983550992, 4.50548970662966015115204524158, 5.72739844306191333795445746318, 6.28909154567699655060071924809, 7.31365724711466279676719255599, 8.036036610785999870763598567055, 8.459076740171270628509899436987, 9.741239634285486443429458139397

Graph of the ZZ-function along the critical line