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Properties

Label 2-3800-5.4-c1-0-78
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  − 1.87i·3-s − 1.18i·7-s − 0.532·9-s − 2.18·11-s − 1.71i·13-s + 0.120i·17-s − 19-s − 2.22·21-s + 7.98i·23-s − 4.63i·27-s − 3.24·29-s − 8.41·31-s + 4.10i·33-s − 3.33i·37-s − 3.22·39-s + ⋯
L(s)  = 1  − 1.08i·3-s − 0.447i·7-s − 0.177·9-s − 0.658·11-s − 0.476i·13-s + 0.0292i·17-s − 0.229·19-s − 0.485·21-s + 1.66i·23-s − 0.892i·27-s − 0.603·29-s − 1.51·31-s + 0.714i·33-s − 0.547i·37-s − 0.516·39-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 0.12708658110.1270865811
L(12)L(\frac12) ≈\approx 0.12708658110.1270865811
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+1.87iT−3T2 1 + 1.87iT - 3T^{2}
7 1+1.18iT−7T2 1 + 1.18iT - 7T^{2}
11 1+2.18T+11T2 1 + 2.18T + 11T^{2}
13 1+1.71iT−13T2 1 + 1.71iT - 13T^{2}
17 1−0.120iT−17T2 1 - 0.120iT - 17T^{2}
23 1−7.98iT−23T2 1 - 7.98iT - 23T^{2}
29 1+3.24T+29T2 1 + 3.24T + 29T^{2}
31 1+8.41T+31T2 1 + 8.41T + 31T^{2}
37 1+3.33iT−37T2 1 + 3.33iT - 37T^{2}
41 1+8.98T+41T2 1 + 8.98T + 41T^{2}
43 1−4.06iT−43T2 1 - 4.06iT - 43T^{2}
47 1−1.71iT−47T2 1 - 1.71iT - 47T^{2}
53 1+6.51iT−53T2 1 + 6.51iT - 53T^{2}
59 1+10.2T+59T2 1 + 10.2T + 59T^{2}
61 1−6.53T+61T2 1 - 6.53T + 61T^{2}
67 1−2.18iT−67T2 1 - 2.18iT - 67T^{2}
71 1+9.12T+71T2 1 + 9.12T + 71T^{2}
73 1−0.773iT−73T2 1 - 0.773iT - 73T^{2}
79 1+1.63T+79T2 1 + 1.63T + 79T^{2}
83 1−2.44iT−83T2 1 - 2.44iT - 83T^{2}
89 1+2.83T+89T2 1 + 2.83T + 89T^{2}
97 1−2.19iT−97T2 1 - 2.19iT - 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

−7.68930783344109921200700207694, −7.43817485568618181143280263689, −6.70816999770833889191373815655, −5.77132493892717145213010733259, −5.20205424644797917296673119903, −4.03646227151723195465242112445, −3.22035561157570939371629440643, −2.09747928391501751434868957842, −1.32219762162599696935788915247, −0.03559923574072089856798902916, 1.79172157845969179923588481772, 2.78471797770305866376345666209, 3.71819956131482776892507752241, 4.47249596264246437902778712816, 5.12282393664470916913127980416, 5.85036417657808291843218844911, 6.79669963382393703493986802937, 7.52252832826002017083922685909, 8.569290555695769117846639102555, 8.942681669265276737220300684282

Graph of the ZZ-function along the critical line