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Properties

Label 2-3800-5.4-c1-0-21
Degree 22
Conductor 38003800
Sign 0.447−0.894i0.447 - 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  − 0.486i·3-s + 3.63i·7-s + 2.76·9-s − 2.79·11-s − 2.86i·13-s + 1.17i·17-s − 19-s + 1.76·21-s + 0.617i·23-s − 2.80i·27-s + 4.96·29-s + 0.745·31-s + 1.36i·33-s + 8.23i·37-s − 1.39·39-s + ⋯
L(s)  = 1  − 0.280i·3-s + 1.37i·7-s + 0.921·9-s − 0.842·11-s − 0.794i·13-s + 0.284i·17-s − 0.229·19-s + 0.385·21-s + 0.128i·23-s − 0.539i·27-s + 0.922·29-s + 0.133·31-s + 0.236i·33-s + 1.35i·37-s − 0.223·39-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: 0.447−0.894i0.447 - 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 1.7236325721.723632572
L(12)L(\frac12) ≈\approx 1.7236325721.723632572
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+0.486iT−3T2 1 + 0.486iT - 3T^{2}
7 1−3.63iT−7T2 1 - 3.63iT - 7T^{2}
11 1+2.79T+11T2 1 + 2.79T + 11T^{2}
13 1+2.86iT−13T2 1 + 2.86iT - 13T^{2}
17 1−1.17iT−17T2 1 - 1.17iT - 17T^{2}
23 1−0.617iT−23T2 1 - 0.617iT - 23T^{2}
29 1−4.96T+29T2 1 - 4.96T + 29T^{2}
31 1−0.745T+31T2 1 - 0.745T + 31T^{2}
37 1−8.23iT−37T2 1 - 8.23iT - 37T^{2}
41 1−9.98T+41T2 1 - 9.98T + 41T^{2}
43 1+10.4iT−43T2 1 + 10.4iT - 43T^{2}
47 1−5.07iT−47T2 1 - 5.07iT - 47T^{2}
53 1−7.45iT−53T2 1 - 7.45iT - 53T^{2}
59 1+3.83T+59T2 1 + 3.83T + 59T^{2}
61 1−11.2T+61T2 1 - 11.2T + 61T^{2}
67 1−6.10iT−67T2 1 - 6.10iT - 67T^{2}
71 1+9.40T+71T2 1 + 9.40T + 71T^{2}
73 1−9.52iT−73T2 1 - 9.52iT - 73T^{2}
79 1−3.70T+79T2 1 - 3.70T + 79T^{2}
83 1−4.66iT−83T2 1 - 4.66iT - 83T^{2}
89 1+10.6T+89T2 1 + 10.6T + 89T^{2}
97 1−0.629iT−97T2 1 - 0.629iT - 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

−8.505275648169241149557266855510, −7.978399692635477727106517732481, −7.22200227210071217793561255222, −6.33295002879587551071303461953, −5.65840528633260754015762968515, −4.99440419626286979827565832039, −4.07535121098109295322587538063, −2.86076881234757663339605883219, −2.32453926498949642698190028562, −1.09840946982056016654984022510, 0.56813902125953105382665538383, 1.71796565932186849500069916082, 2.87699558615811092931288419089, 4.00289113695906207005620083935, 4.38150076578341967162192774991, 5.17270831461869661792096266459, 6.30496450022956417834707552311, 7.01680772205473941745298691743, 7.53781232072267666847099064965, 8.234325372952187047085370066715

Graph of the ZZ-function along the critical line