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Properties

Label 2-756-63.47-c1-0-1
Degree $2$
Conductor $756$
Sign $0.601 - 0.799i$
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  − 2.86·5-s + (−1.83 − 1.90i)7-s + 2.71i·11-s + (3.18 − 1.84i)13-s + (3.22 + 5.58i)17-s + (2.73 + 1.58i)19-s + 2.99i·23-s + 3.22·25-s + (2.48 + 1.43i)29-s + (8.26 + 4.77i)31-s + (5.25 + 5.47i)35-s + (−1.70 + 2.95i)37-s + (−0.794 − 1.37i)41-s + (−4.67 + 8.10i)43-s + (−5.65 − 9.79i)47-s + ⋯
L(s)  = 1  − 1.28·5-s + (−0.692 − 0.721i)7-s + 0.817i·11-s + (0.884 − 0.510i)13-s + (0.781 + 1.35i)17-s + (0.628 + 0.362i)19-s + 0.623i·23-s + 0.645·25-s + (0.461 + 0.266i)29-s + (1.48 + 0.857i)31-s + (0.888 + 0.925i)35-s + (−0.280 + 0.485i)37-s + (−0.124 − 0.214i)41-s + (−0.713 + 1.23i)43-s + (−0.824 − 1.42i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.601 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(756\)    =    \(2^{2} \cdot 3^{3} \cdot 7\)
Sign: $0.601 - 0.799i$
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: \((2,\ 756,\ (\ :1/2),\ 0.601 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.900323 + 0.449258i\)
\(L(\frac12)\) \(\approx\) \(0.900323 + 0.449258i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 + 1.90i)T \)
good5 \( 1 + 2.86T + 5T^{2} \)
11 \( 1 - 2.71iT - 11T^{2} \)
13 \( 1 + (-3.18 + 1.84i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.22 - 5.58i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.73 - 1.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 2.99iT - 23T^{2} \)
29 \( 1 + (-2.48 - 1.43i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.26 - 4.77i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.70 - 2.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.794 + 1.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.67 - 8.10i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.65 + 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.16 + 1.24i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.33 + 7.51i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.566 - 0.327i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.86 - 6.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.86iT - 71T^{2} \)
73 \( 1 + (-11.0 + 6.39i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.59 + 4.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.92 - 13.7i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.14 + 5.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.2 - 7.62i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.28576061937956209417730692222, −9.938371083348867915683235752630, −8.465283814205849951971031618187, −7.960149174487384184652810205365, −7.07485371678458528959213034847, −6.22546481355543225856872952283, −4.91693721124835378329675562742, −3.77641283421246518518778286947, −3.32820477706468656530022368407, −1.20115760396432071213970645928, 0.62014775474290427154392932443, 2.81024646475594682299833701443, 3.56705688467545148752006729499, 4.69408031070244669997440369944, 5.85140062128288496670205924301, 6.72615791003422042138445378846, 7.72370424913677476000762757568, 8.505801546398406782736747558615, 9.234159930940108965741137673411, 10.19206728811697119095561226774

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