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Properties

Label 2-756-28.27-c1-0-58
Degree $2$
Conductor $756$
Sign $-1$
Analytic cond. $6.03669$
Root an. cond. $2.45696$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s − 2.00·4-s − 3.99i·5-s + 2.64·7-s + 2.82i·8-s − 5.64·10-s − 6.31i·11-s − 3.74i·14-s + 4.00·16-s + 3.74i·17-s + 3.35·19-s + 7.98i·20-s − 8.93·22-s − 2.07i·23-s − 10.9·25-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s − 1.78i·5-s + 0.999·7-s + 1.00i·8-s − 1.78·10-s − 1.90i·11-s − 1.00i·14-s + 1.00·16-s + 0.907i·17-s + 0.769·19-s + 1.78i·20-s − 1.90·22-s − 0.433i·23-s − 2.18·25-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $-1$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39004i\)
\(L(\frac12)\) \(\approx\) \(1.39004i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
7 \( 1 - 2.64T \)
good5 \( 1 + 3.99iT - 5T^{2} \)
11 \( 1 + 6.31iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 3.74iT - 17T^{2} \)
19 \( 1 - 3.35T + 19T^{2} \)
23 \( 1 + 2.07iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 2.29T + 31T^{2} \)
37 \( 1 + 11.9T + 37T^{2} \)
41 \( 1 - 8.73iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 2.16iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 7.23iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.963837298908684378077154306899, −8.818377310953809135257461726591, −8.556749804128470095427368672111, −7.926848543245753754650287348066, −5.87564668180646150953077152587, −5.20216638715895349612941496002, −4.39750783513247884967071630298, −3.34880070376196398890156903592, −1.66729105590238308802836333247, −0.77113942865948026636929134961, 2.05995356233578782898175657840, 3.46423420805282078299207995876, 4.63618959596965044482755173777, 5.47946503328480549862527402453, 6.75084894007339945859251442541, 7.28229099466950122468372757604, 7.69607250580846878309019555054, 9.030323121508074707146461683390, 9.975695778439076149749180818320, 10.47848786955036938700431536161

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