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Properties

Label 2-1900-19.18-c0-0-1
Degree $2$
Conductor $1900$
Sign $1$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 9-s − 11-s + 17-s + 19-s − 2·23-s + 43-s + 47-s − 61-s + 63-s + 73-s − 77-s + 81-s − 2·83-s − 99-s + 2·101-s + 119-s + ⋯
L(s)  = 1  + 7-s + 9-s − 11-s + 17-s + 19-s − 2·23-s + 43-s + 47-s − 61-s + 63-s + 73-s − 77-s + 81-s − 2·83-s − 99-s + 2·101-s + 119-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1900} (1101, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.340119955\)
\(L(\frac12)\) \(\approx\) \(1.340119955\)
\(L(1)\) 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 \( ( 1 - T )( 1 + T ) \)
7 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 - T + T^{2} \)
23 \( ( 1 + T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 - T + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 + T )^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.582961294720184443161552074725, −8.446110377620940188979496725684, −7.64140821711644818879550141508, −7.47225143972169562777327167896, −6.08581307066269365216040598307, −5.33965646141524008240387525965, −4.54790916227581866158521276402, −3.65343682061870927329442026144, −2.37941265958342105230705797667, −1.32470092510713737382210346631, 1.32470092510713737382210346631, 2.37941265958342105230705797667, 3.65343682061870927329442026144, 4.54790916227581866158521276402, 5.33965646141524008240387525965, 6.08581307066269365216040598307, 7.47225143972169562777327167896, 7.64140821711644818879550141508, 8.446110377620940188979496725684, 9.582961294720184443161552074725

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