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Properties

Label 2-2736-57.8-c1-0-13
Degree $2$
Conductor $2736$
Sign $0.862 - 0.505i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 - 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.862 - 0.505i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.862 - 0.505i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.768861917\)
\(L(\frac12)\) \(\approx\) \(1.768861917\)
\(L(\frac{3}{2})\) 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

−8.790543383577036968116361748017, −8.065058966049169423362842827279, −7.49887278587292587061289927692, −6.73662829814412114330103823292, −5.70259653780051931982944069258, −4.84080080734335670667118771458, −4.32983034238420952377733368069, −3.23177246123899324822148374392, −2.13055911760924055953111197245, −1.02021719745757179336912921535, 0.71530329685355441995805076448, 2.00133534158613566355148816473, 3.18782583652266441314307416507, 3.83653171616655560490796533368, 4.99337945208420125262107218706, 5.53419053106771853876892375268, 6.49046590620215746433447761581, 7.47917650278083726928177183016, 7.936199990992761971768866483731, 8.554425907887500498712744522357

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