Properties

Label 2-1900-1.1-c1-0-17
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.41·3-s − 0.828·7-s + 8.65·9-s − 2·11-s + 6.24·13-s − 0.828·17-s − 19-s − 2.82·21-s + 6·23-s + 19.3·27-s − 6.48·29-s − 6.82·31-s − 6.82·33-s + 1.75·37-s + 21.3·39-s + 3.65·41-s − 4.82·43-s + 4.82·47-s − 6.31·49-s − 2.82·51-s − 9.07·53-s − 3.41·57-s + 13.6·59-s − 13.6·61-s − 7.17·63-s + 3.41·67-s + 20.4·69-s + ⋯
L(s)  = 1  + 1.97·3-s − 0.313·7-s + 2.88·9-s − 0.603·11-s + 1.73·13-s − 0.200·17-s − 0.229·19-s − 0.617·21-s + 1.25·23-s + 3.71·27-s − 1.20·29-s − 1.22·31-s − 1.18·33-s + 0.288·37-s + 3.41·39-s + 0.571·41-s − 0.736·43-s + 0.704·47-s − 0.901·49-s − 0.396·51-s − 1.24·53-s − 0.452·57-s + 1.77·59-s − 1.74·61-s − 0.903·63-s + 0.417·67-s + 2.46·69-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.729598454\)
\(L(\frac12)\) \(\approx\) \(3.729598454\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.41T + 3T^{2} \)
7 \( 1 + 0.828T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 6.24T + 13T^{2} \)
17 \( 1 + 0.828T + 17T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 + 6.82T + 31T^{2} \)
37 \( 1 - 1.75T + 37T^{2} \)
41 \( 1 - 3.65T + 41T^{2} \)
43 \( 1 + 4.82T + 43T^{2} \)
47 \( 1 - 4.82T + 47T^{2} \)
53 \( 1 + 9.07T + 53T^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 - 3.41T + 67T^{2} \)
71 \( 1 - 5.17T + 71T^{2} \)
73 \( 1 - 2.48T + 73T^{2} \)
79 \( 1 - 1.65T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 + 6.48T + 89T^{2} \)
97 \( 1 - 10.2T + 97T^{2} \)
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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

−9.105350401243179652686633724661, −8.518153926659759368142620055655, −7.82274375813984079578170040733, −7.12570631103319291875353444803, −6.22801289309547539997526798333, −4.95184551981428610559216145997, −3.80120206116264116607526388645, −3.39063017175449050963740697219, −2.38876468739221003337205171349, −1.38357515986915209515578547703, 1.38357515986915209515578547703, 2.38876468739221003337205171349, 3.39063017175449050963740697219, 3.80120206116264116607526388645, 4.95184551981428610559216145997, 6.22801289309547539997526798333, 7.12570631103319291875353444803, 7.82274375813984079578170040733, 8.518153926659759368142620055655, 9.105350401243179652686633724661

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