Properties

Label 2-1850-5.4-c1-0-53
Degree $2$
Conductor $1850$
Sign $0.447 - 0.894i$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(14.7723\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1850} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8836342355\)
\(L(\frac12)\) \(\approx\) \(0.8836342355\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
37 \( 1 - iT \)
good3 \( 1 + 3.30iT - 3T^{2} \)
7 \( 1 + 2.60iT - 7T^{2} \)
11 \( 1 + 2.30T + 11T^{2} \)
13 \( 1 + 1.30iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 3.90iT - 23T^{2} \)
29 \( 1 - 3.90T + 29T^{2} \)
31 \( 1 + 0.302T + 31T^{2} \)
41 \( 1 - 9.90T + 41T^{2} \)
43 \( 1 + 0.605iT - 43T^{2} \)
47 \( 1 - 4.60iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 - 7.51T + 61T^{2} \)
67 \( 1 + 3.51iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 12.3iT - 73T^{2} \)
79 \( 1 + 9.11T + 79T^{2} \)
83 \( 1 + 2.78iT - 83T^{2} \)
89 \( 1 - 9.21T + 89T^{2} \)
97 \( 1 + 16.4iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line