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Properties

Label 2-2736-57.50-c1-0-32
Degree $2$
Conductor $2736$
Sign $0.822 + 0.568i$
Analytic cond. $21.8470$
Root an. cond. $4.67408$
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  + (0.406 − 0.234i)5-s + 3.49·7-s − 1.34i·11-s + (4.30 + 2.48i)13-s + (6.31 − 3.64i)17-s + (−1.88 − 3.92i)19-s + (−6.97 − 4.02i)23-s + (−2.38 + 4.13i)25-s + (4.74 − 8.21i)29-s − 3.95i·31-s + (1.42 − 0.820i)35-s − 0.581i·37-s + (0.761 + 1.31i)41-s + (4.63 + 8.02i)43-s + (−1.06 − 0.615i)47-s + ⋯
L(s)  = 1  + (0.181 − 0.105i)5-s + 1.32·7-s − 0.406i·11-s + (1.19 + 0.689i)13-s + (1.53 − 0.884i)17-s + (−0.433 − 0.901i)19-s + (−1.45 − 0.839i)23-s + (−0.477 + 0.827i)25-s + (0.880 − 1.52i)29-s − 0.711i·31-s + (0.240 − 0.138i)35-s − 0.0955i·37-s + (0.118 + 0.205i)41-s + (0.706 + 1.22i)43-s + (−0.155 − 0.0898i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.822 + 0.568i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.822 + 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.409440936\)
\(L(\frac12)\) \(\approx\) \(2.409440936\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 + (1.88 + 3.92i)T \)
good5 \( 1 + (-0.406 + 0.234i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.49T + 7T^{2} \)
11 \( 1 + 1.34iT - 11T^{2} \)
13 \( 1 + (-4.30 - 2.48i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.31 + 3.64i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (6.97 + 4.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.74 + 8.21i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.95iT - 31T^{2} \)
37 \( 1 + 0.581iT - 37T^{2} \)
41 \( 1 + (-0.761 - 1.31i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.63 - 8.02i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.06 + 0.615i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.09 - 8.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.09 - 8.82i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.59 - 6.22i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.23 + 5.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.67 + 6.36i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.703 + 1.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.56 + 0.903i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.53iT - 83T^{2} \)
89 \( 1 + (-7.73 + 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.66 + 5.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.714859452632312497441315117776, −7.922523048990634944081270666714, −7.53297128468755433906790746174, −6.17155800821594787039065741898, −5.86145955325902607478769013441, −4.67125377674100016130433947466, −4.20590698081972367131327203067, −2.97329883788257369526333804676, −1.89379480222884737511865312672, −0.897022383229640533683284517302, 1.29267010394209246669566996616, 1.92106268246272064653207825526, 3.40794641829269881240026393223, 4.02255925261570367275756500421, 5.16628630401178688578314541791, 5.72457029930336378503840156039, 6.49736216838886776032912386133, 7.68330466379073034544295993345, 8.145909154826410358060706623240, 8.579731794726506719212083863446

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