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Properties

Label 2-2736-57.50-c1-0-14
Degree $2$
Conductor $2736$
Sign $0.189 - 0.981i$
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  + (0.967 − 0.558i)5-s + 1.95·7-s + 2.27i·11-s + (−1.53 − 0.884i)13-s + (−3.94 + 2.27i)17-s + (2.17 + 3.77i)19-s + (0.852 + 0.492i)23-s + (−1.87 + 3.25i)25-s + (−3.73 + 6.47i)29-s + 5.02i·31-s + (1.88 − 1.08i)35-s − 6.12i·37-s + (1.26 + 2.19i)41-s + (1.64 + 2.85i)43-s + (1.16 + 0.675i)47-s + ⋯
L(s)  = 1  + (0.432 − 0.249i)5-s + 0.737·7-s + 0.684i·11-s + (−0.424 − 0.245i)13-s + (−0.956 + 0.552i)17-s + (0.499 + 0.866i)19-s + (0.177 + 0.102i)23-s + (−0.375 + 0.650i)25-s + (−0.693 + 1.20i)29-s + 0.901i·31-s + (0.318 − 0.184i)35-s − 1.00i·37-s + (0.197 + 0.342i)41-s + (0.251 + 0.434i)43-s + (0.170 + 0.0985i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.189 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.189 - 0.981i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.189 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.687668306\)
\(L(\frac12)\) \(\approx\) \(1.687668306\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 - 3.77i)T \)
good5 \( 1 + (-0.967 + 0.558i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.95T + 7T^{2} \)
11 \( 1 - 2.27iT - 11T^{2} \)
13 \( 1 + (1.53 + 0.884i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.94 - 2.27i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.852 - 0.492i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.73 - 6.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.02iT - 31T^{2} \)
37 \( 1 + 6.12iT - 37T^{2} \)
41 \( 1 + (-1.26 - 2.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.64 - 2.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.16 - 0.675i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.63 - 8.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.40 - 7.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.69 + 9.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.67 - 4.43i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.61 + 9.72i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.554 + 0.961i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.63 + 4.98i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.51iT - 83T^{2} \)
89 \( 1 + (-0.860 + 1.49i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.29 - 1.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.114036895187369964290389784277, −8.183730647123949053073818952281, −7.52597721080362708575642261511, −6.79524561008634855134898862000, −5.76210197853554879362019516588, −5.13589752817960295517691449673, −4.37585590097725324485896960899, −3.36931002848273191854717288962, −2.10512539401111920827536049850, −1.41051502542624664806176877627, 0.53325873565392364165167753124, 2.03283756043240979649639646147, 2.69987296814118439132855083532, 3.95026962642980923846517975193, 4.78183447126687729171574658982, 5.52821552197729327895860983208, 6.41113090313595822893747776344, 7.09299338648413969170698583376, 7.972812784668248047707464738673, 8.607249111178667065075694959344

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