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Properties

Label 2-3800-5.4-c1-0-57
Degree 22
Conductor 38003800
Sign −0.894+0.447i-0.894 + 0.447i
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  − 3.12i·3-s + 1.51i·7-s − 6.76·9-s + 4.24·11-s + 4.15i·13-s + 3.51i·17-s + 19-s + 4.73·21-s − 8.73i·23-s + 11.7i·27-s − 1.45·29-s − 4.96·31-s − 13.2i·33-s − 7.60i·37-s + 12.9·39-s + ⋯
L(s)  = 1  − 1.80i·3-s + 0.572i·7-s − 2.25·9-s + 1.28·11-s + 1.15i·13-s + 0.852i·17-s + 0.229·19-s + 1.03·21-s − 1.82i·23-s + 2.26i·27-s − 0.270·29-s − 0.892·31-s − 2.31i·33-s − 1.25i·37-s + 2.07·39-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 38003800    =    23⋅52⋅192^{3} \cdot 5^{2} \cdot 19
Sign: −0.894+0.447i-0.894 + 0.447i
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), −0.894+0.447i)(2,\ 3800,\ (\ :1/2),\ -0.894 + 0.447i)

Particular Values

L(1)L(1) ≈\approx 1.4594877981.459487798
L(12)L(\frac12) ≈\approx 1.4594877981.459487798
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+3.12iT−3T2 1 + 3.12iT - 3T^{2}
7 1−1.51iT−7T2 1 - 1.51iT - 7T^{2}
11 1−4.24T+11T2 1 - 4.24T + 11T^{2}
13 1−4.15iT−13T2 1 - 4.15iT - 13T^{2}
17 1−3.51iT−17T2 1 - 3.51iT - 17T^{2}
23 1+8.73iT−23T2 1 + 8.73iT - 23T^{2}
29 1+1.45T+29T2 1 + 1.45T + 29T^{2}
31 1+4.96T+31T2 1 + 4.96T + 31T^{2}
37 1+7.60iT−37T2 1 + 7.60iT - 37T^{2}
41 1+9.21T+41T2 1 + 9.21T + 41T^{2}
43 1+8.31iT−43T2 1 + 8.31iT - 43T^{2}
47 1+5.28iT−47T2 1 + 5.28iT - 47T^{2}
53 1−0.155iT−53T2 1 - 0.155iT - 53T^{2}
59 1−2.48T+59T2 1 - 2.48T + 59T^{2}
61 1+4.49T+61T2 1 + 4.49T + 61T^{2}
67 1+7.43iT−67T2 1 + 7.43iT - 67T^{2}
71 1−8.49T+71T2 1 - 8.49T + 71T^{2}
73 1+15.0iT−73T2 1 + 15.0iT - 73T^{2}
79 1−0.310T+79T2 1 - 0.310T + 79T^{2}
83 1+8.96iT−83T2 1 + 8.96iT - 83T^{2}
89 1+0.719T+89T2 1 + 0.719T + 89T^{2}
97 1+17.3iT−97T2 1 + 17.3iT - 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

−8.209311645943471055362800098633, −7.21255618833795909528357174289, −6.77435163943699376179243606848, −6.21390545134538307425292591799, −5.52493737294188245204323589832, −4.30000658686166837764257869129, −3.34144194769584065643569725860, −1.99705044600949965226312078625, −1.84532897939226261473956125606, −0.45307212494283683265168155256, 1.17584901098911734538775764151, 2.90191346162540345111680790215, 3.57585510684838047568963495053, 4.08609229368864350604986600749, 5.08517290111110321924192567687, 5.47219334891452742009069214968, 6.48563097191031575804088776563, 7.43674910045826543919413625320, 8.233767660586758626627529036797, 9.103639385913340344282876946568

Graph of the ZZ-function along the critical line