Properties

Label 2-756-21.2-c0-0-0
Degree $2$
Conductor $756$
Sign $0.895 + 0.444i$
Analytic cond. $0.377293$
Root an. cond. $0.614241$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)7-s + 2·13-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)31-s + (−1 + 1.73i)37-s − 43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)61-s + (−1 − 1.73i)67-s + (0.5 + 0.866i)73-s + (−1 + 1.73i)79-s + (−1 − 1.73i)91-s − 97-s + (−1 + 1.73i)103-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + 2·13-s + (0.5 − 0.866i)19-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)31-s + (−1 + 1.73i)37-s − 43-s + (−0.499 + 0.866i)49-s + (0.5 − 0.866i)61-s + (−1 − 1.73i)67-s + (0.5 + 0.866i)73-s + (−1 + 1.73i)79-s + (−1 − 1.73i)91-s − 97-s + (−1 + 1.73i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.895 + 0.444i$
Analytic conductor: \(0.377293\)
Root analytic conductor: \(0.614241\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :0),\ 0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9725035391\)
\(L(\frac12)\) \(\approx\) \(0.9725035391\)
\(L(1)\) 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 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - 2T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.50220222348165524604662537221, −9.742232412987617180579576569954, −8.699164035819916900282065218251, −8.034532254963809259935478754537, −6.78871695268480433648327619841, −6.34137702286563838020390891255, −5.04877944921169936566024468690, −3.92259897083775955625535401179, −3.10468264816957414902063507855, −1.26918191813805124151530000226, 1.68767077795690034784893235093, 3.18449586044372927020330116682, 4.00545553845051295599186923696, 5.61043948651723752723535650071, 5.96747182206304400887987792285, 7.10171719857339533757056547881, 8.232794775295628069526212849968, 8.870239416831465612952893521537, 9.687532573179586339968001977749, 10.62932871528426230517026056625

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