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Properties

Label 2-756-63.47-c1-0-2
Degree $2$
Conductor $756$
Sign $0.661 - 0.749i$
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  − 2.18·5-s + (2.64 − 0.0736i)7-s + 1.46i·11-s + (−2.92 + 1.69i)13-s + (1.32 + 2.28i)17-s + (6.87 + 3.97i)19-s − 4.00i·23-s − 0.234·25-s + (6.71 + 3.87i)29-s + (0.612 + 0.353i)31-s + (−5.77 + 0.160i)35-s + (1.41 − 2.45i)37-s + (3.74 + 6.48i)41-s + (−1.27 + 2.20i)43-s + (6.27 + 10.8i)47-s + ⋯
L(s)  = 1  − 0.976·5-s + (0.999 − 0.0278i)7-s + 0.441i·11-s + (−0.811 + 0.468i)13-s + (0.320 + 0.555i)17-s + (1.57 + 0.911i)19-s − 0.836i·23-s − 0.0468·25-s + (1.24 + 0.719i)29-s + (0.109 + 0.0634i)31-s + (−0.975 + 0.0271i)35-s + (0.233 − 0.403i)37-s + (0.584 + 1.01i)41-s + (−0.193 + 0.335i)43-s + (0.915 + 1.58i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.661 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22447 + 0.552385i\)
\(L(\frac12)\) \(\approx\) \(1.22447 + 0.552385i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-2.64 + 0.0736i)T \)
good5 \( 1 + 2.18T + 5T^{2} \)
11 \( 1 - 1.46iT - 11T^{2} \)
13 \( 1 + (2.92 - 1.69i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.32 - 2.28i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.87 - 3.97i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.00iT - 23T^{2} \)
29 \( 1 + (-6.71 - 3.87i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.612 - 0.353i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.41 + 2.45i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.74 - 6.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.27 - 2.20i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6.27 - 10.8i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.41 - 1.39i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.71 - 11.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.75 + 3.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.92 - 5.05i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (3.95 - 2.28i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.69 + 8.12i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.70 + 2.95i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.61 + 8.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.38 + 3.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.50841652335721055737457268303, −9.655382681312229391370781486045, −8.591904691949681962792771093315, −7.72810617080916187719062745760, −7.35955049731906344708324358056, −6.00779411515560520293509648424, −4.81858806721493320523093297590, −4.21424521077850619016498292250, −2.90019558037418501928267734817, −1.36377729836066157786738104025, 0.794069878812879076593029071100, 2.61147922165791055736819905116, 3.73256439336225935463533601335, 4.87556786584947454968027409175, 5.50473413571227399792748503552, 7.06257575112657673576932984604, 7.67276872223957088012205065095, 8.317529871841451502787743910683, 9.363893708892667103260440796213, 10.25458138674329658245893785899

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