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Properties

Label 2-756-189.5-c1-0-8
Degree $2$
Conductor $756$
Sign $0.794 - 0.607i$
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  + (−1.61 − 0.623i)3-s + (−0.221 + 1.25i)5-s + (2.55 − 0.670i)7-s + (2.22 + 2.01i)9-s + (−5.67 + 1.00i)11-s + (3.02 − 3.60i)13-s + (1.14 − 1.89i)15-s + 0.342·17-s + 8.19i·19-s + (−4.55 − 0.512i)21-s + (0.332 − 0.396i)23-s + (3.16 + 1.15i)25-s + (−2.33 − 4.64i)27-s + (0.622 + 0.741i)29-s + (1.45 + 3.99i)31-s + ⋯
L(s)  = 1  + (−0.932 − 0.360i)3-s + (−0.0991 + 0.562i)5-s + (0.967 − 0.253i)7-s + (0.740 + 0.671i)9-s + (−1.71 + 0.301i)11-s + (0.839 − 1.00i)13-s + (0.294 − 0.488i)15-s + 0.0829·17-s + 1.87i·19-s + (−0.993 − 0.111i)21-s + (0.0693 − 0.0826i)23-s + (0.633 + 0.230i)25-s + (−0.448 − 0.893i)27-s + (0.115 + 0.137i)29-s + (0.261 + 0.718i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(6.03669\)
Root analytic conductor: \(2.45696\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{756} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 756,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04119 + 0.352730i\)
\(L(\frac12)\) \(\approx\) \(1.04119 + 0.352730i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 + 0.623i)T \)
7 \( 1 + (-2.55 + 0.670i)T \)
good5 \( 1 + (0.221 - 1.25i)T + (-4.69 - 1.71i)T^{2} \)
11 \( 1 + (5.67 - 1.00i)T + (10.3 - 3.76i)T^{2} \)
13 \( 1 + (-3.02 + 3.60i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 - 0.342T + 17T^{2} \)
19 \( 1 - 8.19iT - 19T^{2} \)
23 \( 1 + (-0.332 + 0.396i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.622 - 0.741i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.45 - 3.99i)T + (-23.7 + 19.9i)T^{2} \)
37 \( 1 + (1.18 + 2.05i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.78 - 6.52i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-4.17 - 1.52i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-2.92 - 1.06i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (-5.32 + 3.07i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.66 - 8.10i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (2.33 - 6.40i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (0.683 - 3.87i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (2.19 + 1.26i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-10.4 - 6.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.898 + 5.09i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (0.555 - 0.466i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (3.14 - 8.62i)T + (-74.3 - 62.3i)T^{2} \)
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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

−10.64334087804495818762332415471, −10.07672590643213606264910162247, −8.297566681386080093817226348421, −7.83141193461844905696793703614, −7.05302782237806404496663950126, −5.77464428433915630228262304133, −5.33519890594909023993020612744, −4.14443692224744352361440859279, −2.68443297889261204776175803803, −1.22143956298270880666600586550, 0.75094915112844420413457764674, 2.42748913917342531963147013232, 4.15015930912724871472285507062, 4.95554220883661138048377092821, 5.51931723200472887719124637058, 6.66356506898670621308441909987, 7.68785767855078382399759298076, 8.656284886069672498789194869345, 9.321057916706129687789198002778, 10.59463836933675201903566522150

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