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Properties

Label 2-3800-5.4-c1-0-80
Degree 22
Conductor 38003800
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 30.343130.3431
Root an. cond. 5.508465.50846
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.49i·3-s − 2.49i·7-s − 3.20·9-s − 3.49·11-s − 0.713i·13-s − 3.91i·17-s + 19-s − 6.20·21-s − 2.20i·23-s + 0.509i·27-s − 0.636·29-s − 2.14·31-s + 8.69i·33-s − 2.06i·37-s − 1.77·39-s + ⋯
L(s)  = 1  − 1.43i·3-s − 0.941i·7-s − 1.06·9-s − 1.05·11-s − 0.197i·13-s − 0.950i·17-s + 0.229·19-s − 1.35·21-s − 0.459i·23-s + 0.0979i·27-s − 0.118·29-s − 0.384·31-s + 1.51i·33-s − 0.339i·37-s − 0.284·39-s + ⋯

Functional equation

Λ(s)=(3800s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3800s/2 ΓC(s+1/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 30.343130.3431
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3800(3649,⋅)\chi_{3800} (3649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3800, ( :1/2), −0.447−0.894i)(2,\ 3800,\ (\ :1/2),\ -0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.77763671970.7776367197
L(12)L(\frac12) ≈\approx 0.77763671970.7776367197
L(32)L(\frac{3}{2}) 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
5 1 1
19 1−T 1 - T
good3 1+2.49iT−3T2 1 + 2.49iT - 3T^{2}
7 1+2.49iT−7T2 1 + 2.49iT - 7T^{2}
11 1+3.49T+11T2 1 + 3.49T + 11T^{2}
13 1+0.713iT−13T2 1 + 0.713iT - 13T^{2}
17 1+3.91iT−17T2 1 + 3.91iT - 17T^{2}
23 1+2.20iT−23T2 1 + 2.20iT - 23T^{2}
29 1+0.636T+29T2 1 + 0.636T + 29T^{2}
31 1+2.14T+31T2 1 + 2.14T + 31T^{2}
37 1+2.06iT−37T2 1 + 2.06iT - 37T^{2}
41 1+4.35T+41T2 1 + 4.35T + 41T^{2}
43 1+4.55iT−43T2 1 + 4.55iT - 43T^{2}
47 1+0.268iT−47T2 1 + 0.268iT - 47T^{2}
53 1−7.98iT−53T2 1 - 7.98iT - 53T^{2}
59 1−2.57T+59T2 1 - 2.57T + 59T^{2}
61 1+10.8T+61T2 1 + 10.8T + 61T^{2}
67 1−1.08iT−67T2 1 - 1.08iT - 67T^{2}
71 1−6.83T+71T2 1 - 6.83T + 71T^{2}
73 1−15.0iT−73T2 1 - 15.0iT - 73T^{2}
79 1−7.63T+79T2 1 - 7.63T + 79T^{2}
83 1−0.923iT−83T2 1 - 0.923iT - 83T^{2}
89 1+14.1T+89T2 1 + 14.1T + 89T^{2}
97 1+6.14iT−97T2 1 + 6.14iT - 97T^{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

−7.82257974461504780548865685208, −7.21125607724595925161636679169, −6.86530907499389929677450502919, −5.85159655395787988159237126191, −5.13800651578231756219686026568, −4.14169948379037910479312859877, −3.00900683175834972874577745819, −2.25296497467054431748179219903, −1.14088560416699761886137482040, −0.23519039623746656479886125353, 1.83616106002060038410262931476, 2.90061353124348997683568267497, 3.60662581388110246341105251411, 4.50899746438590120917984078089, 5.24321224224846928409430435005, 5.70059397161161435704024811313, 6.63980853911640696104331333460, 7.79288666685536375576513616230, 8.374676896734722441570468794020, 9.161579232077970635865191661558

Graph of the ZZ-function along the critical line