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Properties

Label 2-756-28.27-c1-0-47
Degree $2$
Conductor $756$
Sign $0.395 + 0.918i$
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  + (1.15 − 0.818i)2-s + (0.660 − 1.88i)4-s + 2.16i·5-s + (1.94 − 1.79i)7-s + (−0.782 − 2.71i)8-s + (1.76 + 2.49i)10-s − 0.414i·11-s − 2.36i·13-s + (0.782 − 3.65i)14-s + (−3.12 − 2.49i)16-s − 0.695i·17-s + 5.21·19-s + (4.08 + 1.42i)20-s + (−0.339 − 0.478i)22-s + 0.414i·23-s + ⋯
L(s)  = 1  + (0.815 − 0.578i)2-s + (0.330 − 0.943i)4-s + 0.967i·5-s + (0.736 − 0.676i)7-s + (−0.276 − 0.960i)8-s + (0.559 + 0.788i)10-s − 0.124i·11-s − 0.656i·13-s + (0.209 − 0.977i)14-s + (−0.781 − 0.623i)16-s − 0.168i·17-s + 1.19·19-s + (0.912 + 0.319i)20-s + (−0.0723 − 0.101i)22-s + 0.0864i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.395 + 0.918i$
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: \((2,\ 756,\ (\ :1/2),\ 0.395 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18249 - 1.43683i\)
\(L(\frac12)\) \(\approx\) \(2.18249 - 1.43683i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 0.818i)T \)
3 \( 1 \)
7 \( 1 + (-1.94 + 1.79i)T \)
good5 \( 1 - 2.16iT - 5T^{2} \)
11 \( 1 + 0.414iT - 11T^{2} \)
13 \( 1 + 2.36iT - 13T^{2} \)
17 \( 1 + 0.695iT - 17T^{2} \)
19 \( 1 - 5.21T + 19T^{2} \)
23 \( 1 - 0.414iT - 23T^{2} \)
29 \( 1 - 9.73T + 29T^{2} \)
31 \( 1 + 2.03T + 31T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 - 10.5iT - 41T^{2} \)
43 \( 1 + 8.76iT - 43T^{2} \)
47 \( 1 + 5.11T + 47T^{2} \)
53 \( 1 - 3.12T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 - 3.97iT - 61T^{2} \)
67 \( 1 + 2.36iT - 67T^{2} \)
71 \( 1 - 4.04iT - 71T^{2} \)
73 \( 1 - 12.3iT - 73T^{2} \)
79 \( 1 - 7.61iT - 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 + 3.97iT - 97T^{2} \)
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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

−10.39348252544085791004781907343, −9.810239650276179980024040885505, −8.413924530670719636917125526922, −7.31684647331367963057766345114, −6.67848058726480090425829884858, −5.51741112154044923965466695430, −4.68860640349910671789472617297, −3.50712056423131877847108422611, −2.73316976755688573725253221814, −1.20462648073589392570416876371, 1.70177444763141456543551710998, 3.11331484324358116240431673539, 4.50145746333776382335419151048, 5.00785368479160170449716575430, 5.87823386555576700813810877775, 6.94626849229568317716031085858, 7.927826859001230875278532015879, 8.658588499015753657283865301101, 9.287524575976850977070169669847, 10.68593898800873536906772087346

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