Properties

Label 2-1900-5.4-c1-0-3
Degree $2$
Conductor $1900$
Sign $-0.447 - 0.894i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
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  − 0.585i·3-s + 4.82i·7-s + 2.65·9-s − 2·11-s + 2.24i·13-s + 4.82i·17-s + 19-s + 2.82·21-s − 6i·23-s − 3.31i·27-s − 10.4·29-s − 1.17·31-s + 1.17i·33-s + 10.2i·37-s + 1.31·39-s + ⋯
L(s)  = 1  − 0.338i·3-s + 1.82i·7-s + 0.885·9-s − 0.603·11-s + 0.621i·13-s + 1.17i·17-s + 0.229·19-s + 0.617·21-s − 1.25i·23-s − 0.637i·27-s − 1.94·29-s − 0.210·31-s + 0.203i·33-s + 1.68i·37-s + 0.210·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1749, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.240746167\)
\(L(\frac12)\) \(\approx\) \(1.240746167\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + 0.585iT - 3T^{2} \)
7 \( 1 - 4.82iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 2.24iT - 13T^{2} \)
17 \( 1 - 4.82iT - 17T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + 1.17T + 31T^{2} \)
37 \( 1 - 10.2iT - 37T^{2} \)
41 \( 1 + 7.65T + 41T^{2} \)
43 \( 1 + 0.828iT - 43T^{2} \)
47 \( 1 + 0.828iT - 47T^{2} \)
53 \( 1 + 5.07iT - 53T^{2} \)
59 \( 1 + 2.34T + 59T^{2} \)
61 \( 1 + 2.34T + 61T^{2} \)
67 \( 1 - 0.585iT - 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 - 14.4iT - 73T^{2} \)
79 \( 1 - 9.65T + 79T^{2} \)
83 \( 1 - 9.31iT - 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 - 1.75iT - 97T^{2} \)
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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

−9.457567330720657875628495324974, −8.525248525222699455922657116350, −8.140698389547391115969866666685, −7.00113335969234800362936106053, −6.30208408258863143993010291043, −5.51093407788726407953362736484, −4.70788287172161161676475745778, −3.56433024233249576319646014572, −2.35724105000585277455318473274, −1.70306319486223363962787945059, 0.43965891675723827178753030623, 1.69876894568532541943955976164, 3.31064518014614791860158522328, 3.89838532985512108090909406237, 4.83156699223862316259037237170, 5.55695153622371043713285044332, 6.88378941129265558520382301297, 7.53538556137534499880034663050, 7.73755213979600958324637983219, 9.329281422350675336055303414628

Graph of the $Z$-function along the critical line