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Properties

Label 2-3800-152.11-c0-0-5
Degree 22
Conductor 38003800
Sign −0.305+0.952i-0.305 + 0.952i
Analytic cond. 1.896441.89644
Root an. cond. 1.377111.37711
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  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s − 0.999·8-s − 11-s + 0.999·12-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)24-s − 27-s + (0.499 + 0.866i)32-s + (0.5 − 0.866i)33-s + (−0.999 − 1.73i)34-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s − 0.999·8-s − 11-s + 0.999·12-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)24-s − 27-s + (0.499 + 0.866i)32-s + (0.5 − 0.866i)33-s + (−0.999 − 1.73i)34-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.305+0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.305+0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.305+0.952i-0.305 + 0.952i
Analytic conductor: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(3051,⋅)\chi_{3800} (3051, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), −0.305+0.952i)(2,\ 3800,\ (\ :0),\ -0.305 + 0.952i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0053813331.005381333
L(12)L(\frac12) ≈\approx 1.0053813331.005381333
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
5 1 1
19 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good3 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
7 1−T2 1 - T^{2}
11 1+T+T2 1 + T + T^{2}
13 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
17 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1−T2 1 - T^{2}
37 1−T2 1 - T^{2}
41 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
43 1+(−1+1.73i)T+(−0.5−0.866i)T2 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
59 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
61 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
67 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1−T+T2 1 - T + T^{2}
89 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
97 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)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

−8.801061667146172223900006302073, −7.67910070770663830479328246272, −6.94944209964752139845463856012, −5.66467494285632800173711656226, −5.32763178529595750794184896519, −4.67232190865074126792521798506, −3.91585151042771748337942254484, −2.92764953168571422288492782366, −2.20386241958171626950647796076, −0.54689964200041478865156603202, 1.30942933133815676285840403259, 2.61457635491670598605040696977, 3.69640326728159162261851970047, 4.40672017324912073846013203006, 5.58269055062843969766651745156, 5.91220156149461080410026856878, 6.53633365398466010734190753223, 7.46696163752520357293982429056, 7.908190608243251120909372638082, 8.455486710762704136513495198753

Graph of the ZZ-function along the critical line