Properties

Label 2-1900-1.1-c1-0-13
Degree $2$
Conductor $1900$
Sign $-1$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.58·3-s − 2.81·7-s + 3.69·9-s + 1.58·11-s − 0.888·13-s + 3.98·17-s + 19-s + 7.27·21-s − 3.30·23-s − 1.81·27-s + 6.98·29-s − 4.51·31-s − 4.11·33-s + 2.41·37-s + 2.30·39-s + 5.09·41-s − 8.17·43-s + 9.08·47-s + 0.901·49-s − 10.3·51-s − 7.79·53-s − 2.58·57-s − 2.22·59-s − 9.90·61-s − 10.3·63-s + 7.56·67-s + 8.54·69-s + ⋯
L(s)  = 1  − 1.49·3-s − 1.06·7-s + 1.23·9-s + 0.478·11-s − 0.246·13-s + 0.967·17-s + 0.229·19-s + 1.58·21-s − 0.688·23-s − 0.348·27-s + 1.29·29-s − 0.810·31-s − 0.715·33-s + 0.396·37-s + 0.368·39-s + 0.796·41-s − 1.24·43-s + 1.32·47-s + 0.128·49-s − 1.44·51-s − 1.07·53-s − 0.342·57-s − 0.289·59-s − 1.26·61-s − 1.31·63-s + 0.924·67-s + 1.02·69-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.58T + 3T^{2} \)
7 \( 1 + 2.81T + 7T^{2} \)
11 \( 1 - 1.58T + 11T^{2} \)
13 \( 1 + 0.888T + 13T^{2} \)
17 \( 1 - 3.98T + 17T^{2} \)
23 \( 1 + 3.30T + 23T^{2} \)
29 \( 1 - 6.98T + 29T^{2} \)
31 \( 1 + 4.51T + 31T^{2} \)
37 \( 1 - 2.41T + 37T^{2} \)
41 \( 1 - 5.09T + 41T^{2} \)
43 \( 1 + 8.17T + 43T^{2} \)
47 \( 1 - 9.08T + 47T^{2} \)
53 \( 1 + 7.79T + 53T^{2} \)
59 \( 1 + 2.22T + 59T^{2} \)
61 \( 1 + 9.90T + 61T^{2} \)
67 \( 1 - 7.56T + 67T^{2} \)
71 \( 1 + 0.777T + 71T^{2} \)
73 \( 1 - 0.876T + 73T^{2} \)
79 \( 1 + 8.94T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + 5.27T + 89T^{2} \)
97 \( 1 + 7.24T + 97T^{2} \)
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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

−9.017332873402318905296629812349, −7.85994217001762863168567379537, −6.98150973979863020460703349302, −6.29357708955303672461640857227, −5.77246462337822238244804741764, −4.90639725851899570369214566549, −3.94046108998624592801582996816, −2.89571547232867225181889414913, −1.24315191819665381359586901850, 0, 1.24315191819665381359586901850, 2.89571547232867225181889414913, 3.94046108998624592801582996816, 4.90639725851899570369214566549, 5.77246462337822238244804741764, 6.29357708955303672461640857227, 6.98150973979863020460703349302, 7.85994217001762863168567379537, 9.017332873402318905296629812349

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