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Properties

Label 2-3800-152.35-c0-0-4
Degree 22
Conductor 38003800
Sign 0.756−0.654i0.756 - 0.654i
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  + (0.766 + 0.642i)2-s + (1.76 − 0.642i)3-s + (0.173 + 0.984i)4-s + (1.76 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (1.93 − 1.62i)9-s + (−0.766 + 1.32i)11-s + (0.939 + 1.62i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 2.53·18-s + (0.766 − 0.642i)19-s + (−1.43 + 0.524i)22-s + (−0.326 + 1.85i)24-s + (1.43 − 2.49i)27-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (1.76 − 0.642i)3-s + (0.173 + 0.984i)4-s + (1.76 + 0.642i)6-s + (−0.500 + 0.866i)8-s + (1.93 − 1.62i)9-s + (−0.766 + 1.32i)11-s + (0.939 + 1.62i)12-s + (−0.939 + 0.342i)16-s + (−0.766 − 0.642i)17-s + 2.53·18-s + (0.766 − 0.642i)19-s + (−1.43 + 0.524i)22-s + (−0.326 + 1.85i)24-s + (1.43 − 2.49i)27-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((0.756−0.654i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s) L(s)=((0.756−0.654i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: 0.756−0.654i0.756 - 0.654i
Analytic conductor: 1.896441.89644
Root analytic conductor: 1.377111.37711
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3800(1251,⋅)\chi_{3800} (1251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :0), 0.756−0.654i)(2,\ 3800,\ (\ :0),\ 0.756 - 0.654i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 3.4448985183.444898518
L(12)L(\frac12) ≈\approx 3.4448985183.444898518
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+(−0.766−0.642i)T 1 + (-0.766 - 0.642i)T
5 1 1
19 1+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
good3 1+(−1.76+0.642i)T+(0.766−0.642i)T2 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2}
7 1+(0.5−0.866i)T2 1 + (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+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
17 1+(0.766+0.642i)T+(0.173+0.984i)T2 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2}
23 1+(0.939−0.342i)T2 1 + (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−T2 1 - T^{2}
41 1+(−1.76+0.642i)T+(0.766−0.642i)T2 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2}
43 1+(0.173−0.984i)T+(−0.939−0.342i)T2 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}
47 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
53 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
59 1+(1.43+1.20i)T+(0.173+0.984i)T2 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2}
61 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
67 1+(1.43−1.20i)T+(0.173−0.984i)T2 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2}
71 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
73 1+(0.326−0.118i)T+(0.766−0.642i)T2 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2}
79 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
83 1+(0.766+1.32i)T+(−0.5+0.866i)T2 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2}
89 1+(1.87+0.684i)T+(0.766+0.642i)T2 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2}
97 1+(−0.266−0.223i)T+(0.173+0.984i)T2 1 + (-0.266 - 0.223i)T + (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

−8.536335493528448860733247459210, −7.79994507604771304407138163382, −7.29782506637038413920829691573, −6.95176513340206908944139673891, −5.90352935693784931784445286000, −4.67145421097925607364092112379, −4.30842202658855752992176899649, −3.05293389421121208712818099326, −2.68580524566222964331829090725, −1.77345951636550670047352189157, 1.50786051014669123445812855798, 2.54126919979046509345881878895, 3.09218346791879366697914252569, 3.79334936878745012956437526042, 4.45093724303031218212184007107, 5.37520621460599890396709574141, 6.16459398717167379971306559774, 7.31857728208881232088348872629, 8.058930693361989386536622887179, 8.700061087682479727104865563801

Graph of the ZZ-function along the critical line