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Properties

Label 2-1850-5.4-c1-0-53
Degree 22
Conductor 18501850
Sign 0.447−0.894i0.447 - 0.894i
Analytic cond. 14.772314.7723
Root an. cond. 3.843473.84347
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  − i·2-s − 3.30i·3-s − 4-s − 3.30·6-s − 2.60i·7-s + i·8-s − 7.90·9-s − 2.30·11-s + 3.30i·12-s − 1.30i·13-s − 2.60·14-s + 16-s − 6i·17-s + 7.90i·18-s − 2·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.90i·3-s − 0.5·4-s − 1.34·6-s − 0.984i·7-s + 0.353i·8-s − 2.63·9-s − 0.694·11-s + 0.953i·12-s − 0.361i·13-s − 0.696·14-s + 0.250·16-s − 1.45i·17-s + 1.86i·18-s − 0.458·19-s + ⋯

Functional equation

Λ(s)=(1850s/2 ΓC(s) L(s)=((0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1850s/2 ΓC(s+1/2) L(s)=((0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{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: 18501850    =    2⋅52⋅372 \cdot 5^{2} \cdot 37
Sign: 0.447−0.894i0.447 - 0.894i
Analytic conductor: 14.772314.7723
Root analytic conductor: 3.843473.84347
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1850(149,⋅)\chi_{1850} (149, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1850, ( :1/2), 0.447−0.894i)(2,\ 1850,\ (\ :1/2),\ 0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 0.88363423550.8836342355
L(12)L(\frac12) ≈\approx 0.88363423550.8836342355
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+iT 1 + iT
5 1 1
37 1−iT 1 - iT
good3 1+3.30iT−3T2 1 + 3.30iT - 3T^{2}
7 1+2.60iT−7T2 1 + 2.60iT - 7T^{2}
11 1+2.30T+11T2 1 + 2.30T + 11T^{2}
13 1+1.30iT−13T2 1 + 1.30iT - 13T^{2}
17 1+6iT−17T2 1 + 6iT - 17T^{2}
19 1+2T+19T2 1 + 2T + 19T^{2}
23 1+3.90iT−23T2 1 + 3.90iT - 23T^{2}
29 1−3.90T+29T2 1 - 3.90T + 29T^{2}
31 1+0.302T+31T2 1 + 0.302T + 31T^{2}
41 1−9.90T+41T2 1 - 9.90T + 41T^{2}
43 1+0.605iT−43T2 1 + 0.605iT - 43T^{2}
47 1−4.60iT−47T2 1 - 4.60iT - 47T^{2}
53 1−6iT−53T2 1 - 6iT - 53T^{2}
59 1+10.6T+59T2 1 + 10.6T + 59T^{2}
61 1−7.51T+61T2 1 - 7.51T + 61T^{2}
67 1+3.51iT−67T2 1 + 3.51iT - 67T^{2}
71 1−6T+71T2 1 - 6T + 71T^{2}
73 1−12.3iT−73T2 1 - 12.3iT - 73T^{2}
79 1+9.11T+79T2 1 + 9.11T + 79T^{2}
83 1+2.78iT−83T2 1 + 2.78iT - 83T^{2}
89 1−9.21T+89T2 1 - 9.21T + 89T^{2}
97 1+16.4iT−97T2 1 + 16.4iT - 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

−8.401886326933767201963264357225, −7.72135383188643366673300531933, −7.18532353862061470008727743972, −6.37916493416222010322830762330, −5.43602758158377087054906794543, −4.40597362503277572973340579017, −2.95965143476785906450790399701, −2.43925672926308053463527222283, −1.14046419582047499651588051164, −0.35897365762615882054820879967, 2.39386884788498814711500129560, 3.50602512263894727306296580840, 4.26247640333869065064156096916, 5.09717352272085521908028714941, 5.71868154862360284281038166984, 6.35398749140177310297830914921, 7.84454553961700542486236218614, 8.527273918355052946531069953506, 9.067297388263805762324121347360, 9.750713744738748517869496790594

Graph of the ZZ-function along the critical line