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Properties

Label 2-2736-76.27-c1-0-21
Degree $2$
Conductor $2736$
Sign $0.938 - 0.345i$
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  + (0.767 − 1.32i)5-s − 2.31i·7-s + 5.49i·11-s + (−3.96 + 2.29i)13-s + (2.79 − 4.84i)17-s + (1.09 + 4.21i)19-s + (3.07 − 1.77i)23-s + (1.32 + 2.29i)25-s + (−7.12 + 4.11i)29-s + 6.20·31-s + (−3.07 − 1.77i)35-s + 1.73i·37-s + (5.86 + 3.38i)41-s + (9.30 + 5.37i)43-s + (−1.68 + 0.973i)47-s + ⋯
L(s)  = 1  + (0.343 − 0.594i)5-s − 0.874i·7-s + 1.65i·11-s + (−1.10 + 0.635i)13-s + (0.678 − 1.17i)17-s + (0.252 + 0.967i)19-s + (0.641 − 0.370i)23-s + (0.264 + 0.458i)25-s + (−1.32 + 0.764i)29-s + 1.11·31-s + (−0.519 − 0.300i)35-s + 0.284i·37-s + (0.916 + 0.528i)41-s + (1.41 + 0.819i)43-s + (−0.246 + 0.142i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.938 - 0.345i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.938 - 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.813039026\)
\(L(\frac12)\) \(\approx\) \(1.813039026\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.09 - 4.21i)T \)
good5 \( 1 + (-0.767 + 1.32i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 2.31iT - 7T^{2} \)
11 \( 1 - 5.49iT - 11T^{2} \)
13 \( 1 + (3.96 - 2.29i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.79 + 4.84i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.07 + 1.77i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.12 - 4.11i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.20T + 31T^{2} \)
37 \( 1 - 1.73iT - 37T^{2} \)
41 \( 1 + (-5.86 - 3.38i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.30 - 5.37i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.68 - 0.973i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.16 - 4.71i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.76 + 8.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.32 + 4.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.20 - 7.27i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.83 - 13.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.79 + 11.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.11 + 12.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.94iT - 83T^{2} \)
89 \( 1 + (4.82 - 2.78i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.40 - 4.27i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.186004338495087688070516258255, −7.85143361550723378580939957222, −7.35926212099111669219553216992, −6.84492193018989433578801656500, −5.64866090049413674879291298775, −4.72037395321948235124519541850, −4.47314478728593411591371746156, −3.15701675091456065551079187624, −2.03235088948591581522467347350, −1.04566339063805615997116649641, 0.69774702160562517015556094323, 2.33949793615190602921892978512, 2.90293785478793546937009164340, 3.82657696210987543366563484153, 5.14133990749251000942902382300, 5.77141882986858392212054231263, 6.27398038138473345502397396710, 7.34222350801149329713344566729, 8.039439800010335067438646239579, 8.819242444008425078193782967336

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