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Properties

Label 2-1872-52.51-c0-0-0
Degree $2$
Conductor $1872$
Sign $1$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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  − 2·7-s + 13-s + 2·19-s + 25-s + 2·31-s + 3·49-s − 2·61-s − 2·67-s − 2·91-s + ⋯
L(s)  = 1  − 2·7-s + 13-s + 2·19-s + 25-s + 2·31-s + 3·49-s − 2·61-s − 2·67-s − 2·91-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1872} (415, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9981779687\)
\(L(\frac12)\) \(\approx\) \(0.9981779687\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 - T \)
good5 \( ( 1 - T )( 1 + T ) \)
7 \( ( 1 + T )^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( ( 1 - T )^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( 1 + T^{2} \)
31 \( ( 1 - T )^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( ( 1 + T )^{2} \)
67 \( ( 1 + T )^{2} \)
71 \( 1 + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
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

−9.405699855998794196490633020512, −8.865265241627971360987530370065, −7.79109794351366990142282425789, −6.91036367986177424592172409604, −6.29036216409153054952156669611, −5.60867477635528805020784324723, −4.40835532617469295766436986151, −3.24388696821516910428269757435, −2.95909812871274582697050335839, −1.04003023657980847128158712422, 1.04003023657980847128158712422, 2.95909812871274582697050335839, 3.24388696821516910428269757435, 4.40835532617469295766436986151, 5.60867477635528805020784324723, 6.29036216409153054952156669611, 6.91036367986177424592172409604, 7.79109794351366990142282425789, 8.865265241627971360987530370065, 9.405699855998794196490633020512

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