Properties

Label 2-756-84.83-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.5332209760\)
\(L(\frac12)\) \(\approx\) \(0.5332209760\)
\(L(1)\) 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

−10.28176671361445161923648708462, −9.709820991530204811583283397928, −8.906051884537861062525416013108, −7.972562304988798816726006679675, −7.24286208203389081154228430855, −6.50147871457115738114872337097, −5.39164668529538021141213699209, −3.66901223687432574881088411564, −3.09776787425856463816338690445, −1.10292190030766665047006066921, 1.10292190030766665047006066921, 3.09776787425856463816338690445, 3.66901223687432574881088411564, 5.39164668529538021141213699209, 6.50147871457115738114872337097, 7.24286208203389081154228430855, 7.972562304988798816726006679675, 8.906051884537861062525416013108, 9.709820991530204811583283397928, 10.28176671361445161923648708462

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