Properties

Label 2-1872-52.51-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.425154569\)
\(L(\frac12)\) \(\approx\) \(1.425154569\)
\(L(1)\) 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

−9.115510199133622270973128704807, −8.578588622745030289198918838024, −8.022043561524797072549103790355, −7.15821272905254465886867572347, −6.19188528264289808706609359906, −5.28945595567866566047615152053, −4.53421400002794327137538132756, −3.76407219602807927901906454278, −2.24756596036987072637898034166, −1.43103210819085659337104044442, 1.43103210819085659337104044442, 2.24756596036987072637898034166, 3.76407219602807927901906454278, 4.53421400002794327137538132756, 5.28945595567866566047615152053, 6.19188528264289808706609359906, 7.15821272905254465886867572347, 8.022043561524797072549103790355, 8.578588622745030289198918838024, 9.115510199133622270973128704807

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