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Properties

Label 2-3800-5.4-c1-0-26
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  + 0.347i·3-s + 3.41i·7-s + 2.87·9-s + 2.41·11-s + 6.29i·13-s + 2.34i·17-s − 19-s − 1.18·21-s + 2.49i·23-s + 2.04i·27-s + 8.17·29-s − 2.77·31-s + 0.837i·33-s + 0.977i·37-s − 2.18·39-s + ⋯
L(s)  = 1  + 0.200i·3-s + 1.28i·7-s + 0.959·9-s + 0.727·11-s + 1.74i·13-s + 0.569i·17-s − 0.229·19-s − 0.258·21-s + 0.519i·23-s + 0.392i·27-s + 1.51·29-s − 0.498·31-s + 0.145i·33-s + 0.160i·37-s − 0.349·39-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 1.9893297241.989329724
L(12)L(\frac12) ≈\approx 1.9893297241.989329724
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 1−0.347iT−3T2 1 - 0.347iT - 3T^{2}
7 1−3.41iT−7T2 1 - 3.41iT - 7T^{2}
11 1−2.41T+11T2 1 - 2.41T + 11T^{2}
13 1−6.29iT−13T2 1 - 6.29iT - 13T^{2}
17 1−2.34iT−17T2 1 - 2.34iT - 17T^{2}
23 1−2.49iT−23T2 1 - 2.49iT - 23T^{2}
29 1−8.17T+29T2 1 - 8.17T + 29T^{2}
31 1+2.77T+31T2 1 + 2.77T + 31T^{2}
37 1−0.977iT−37T2 1 - 0.977iT - 37T^{2}
41 1+3.49T+41T2 1 + 3.49T + 41T^{2}
43 1+2.75iT−43T2 1 + 2.75iT - 43T^{2}
47 1+6.29iT−47T2 1 + 6.29iT - 47T^{2}
53 1−2.38iT−53T2 1 - 2.38iT - 53T^{2}
59 1+3.67T+59T2 1 + 3.67T + 59T^{2}
61 1+12.7T+61T2 1 + 12.7T + 61T^{2}
67 1+2.41iT−67T2 1 + 2.41iT - 67T^{2}
71 1−4.51T+71T2 1 - 4.51T + 71T^{2}
73 1−1.81iT−73T2 1 - 1.81iT - 73T^{2}
79 1−5.04T+79T2 1 - 5.04T + 79T^{2}
83 1+8.07iT−83T2 1 + 8.07iT - 83T^{2}
89 1−2.94T+89T2 1 - 2.94T + 89T^{2}
97 1−3.09iT−97T2 1 - 3.09iT - 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

−8.942763432709325135908630866790, −8.154230513918400660110746195226, −7.07356802786268792572702282622, −6.55993858303700047524542450020, −5.85505911803266729765354787691, −4.82532741913189598420109175174, −4.24516276357880618264776846238, −3.36571280805303564520261683484, −2.12171625357295156442403259898, −1.50601740205259182684811960351, 0.62440370078852320082160612569, 1.35528109140582765790626247289, 2.76552139917970842833926793925, 3.65071992631756074727208803121, 4.42115863555398423969183010761, 5.10173832176468083521331896749, 6.27863755806285691844212619052, 6.79034462353144567564949155096, 7.62005158822670028803309901470, 7.975727670159443995970786172481

Graph of the ZZ-function along the critical line