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Properties

Label 2-3192-3192.797-c0-0-23
Degree $2$
Conductor $3192$
Sign $1$
Analytic cond. $1.59301$
Root an. cond. $1.26214$
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-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 12-s − 14-s + 16-s + 18-s − 19-s − 21-s + 24-s + 25-s + 27-s − 28-s + 32-s + 36-s − 38-s − 42-s + 48-s + 49-s + 50-s + 54-s − 56-s − 57-s + ⋯
L(s)  = 1  + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 12-s − 14-s + 16-s + 18-s − 19-s − 21-s + 24-s + 25-s + 27-s − 28-s + 32-s + 36-s − 38-s − 42-s + 48-s + 49-s + 50-s + 54-s − 56-s − 57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3192\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(1.59301\)
Root analytic conductor: \(1.26214\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3192} (797, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3192,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.144043159\)
\(L(\frac12)\) \(\approx\) \(3.144043159\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
7 \( 1 + T \)
19 \( 1 + T \)
good5 \( ( 1 - T )( 1 + T ) \)
11 \( 1 + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
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 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T^{2} \)
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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

−8.871871187572030857875992288122, −7.978908528074377906828360286512, −7.22591376078376709145205018755, −6.57094137252005987736383245516, −5.92682665195104732043960205839, −4.74539400389285829018088217181, −4.11995077500143505516475260619, −3.19785295686533596890691623575, −2.70454960633492071904108972221, −1.59201705226022670499623273205, 1.59201705226022670499623273205, 2.70454960633492071904108972221, 3.19785295686533596890691623575, 4.11995077500143505516475260619, 4.74539400389285829018088217181, 5.92682665195104732043960205839, 6.57094137252005987736383245516, 7.22591376078376709145205018755, 7.978908528074377906828360286512, 8.871871187572030857875992288122

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