Properties

Label 2-756-84.83-c0-0-3
Degree $2$
Conductor $756$
Sign $1$
Analytic cond. $0.377293$
Root an. cond. $0.614241$
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 + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s − 14-s + 16-s − 2·17-s + 19-s + 20-s − 22-s − 23-s − 28-s + 31-s + 32-s − 2·34-s − 35-s − 37-s + 38-s + 40-s + 41-s − 44-s − 46-s + 49-s − 55-s + ⋯
L(s)  = 1  + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 11-s − 14-s + 16-s − 2·17-s + 19-s + 20-s − 22-s − 23-s − 28-s + 31-s + 32-s − 2·34-s − 35-s − 37-s + 38-s + 40-s + 41-s − 44-s − 46-s + 49-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.377293\)
Root analytic conductor: \(0.614241\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{756} (755, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.753521755\)
\(L(\frac12)\) \(\approx\) \(1.753521755\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - T + T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 + T )^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 - T + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( 1 - T + T^{2} \)
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

−10.48860615324319911172915212623, −9.957349023874567555417663990553, −8.984313395998733290831877798221, −7.72215874108065267747623198746, −6.71490057619476191556249838639, −6.07463380538032471166318897736, −5.26800699997401246927775108992, −4.19440565051837915297271177807, −2.91867617249410807579486425175, −2.11132364390146111360862922278, 2.11132364390146111360862922278, 2.91867617249410807579486425175, 4.19440565051837915297271177807, 5.26800699997401246927775108992, 6.07463380538032471166318897736, 6.71490057619476191556249838639, 7.72215874108065267747623198746, 8.984313395998733290831877798221, 9.957349023874567555417663990553, 10.48860615324319911172915212623

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