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Properties

Label 2-2736-76.27-c1-0-12
Degree $2$
Conductor $2736$
Sign $-0.296 - 0.955i$
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  + (0.353 − 0.612i)5-s + 0.0924i·7-s + 4.18i·11-s + (3.54 − 2.04i)13-s + (−1.22 + 2.12i)17-s + (−4.26 + 0.879i)19-s + (−3.10 + 1.79i)23-s + (2.24 + 3.89i)25-s + (−7.24 + 4.18i)29-s − 3.79·31-s + (0.0566 + 0.0326i)35-s − 1.91i·37-s + (−6.72 − 3.88i)41-s + (7.76 + 4.48i)43-s + (2.64 − 1.52i)47-s + ⋯
L(s)  = 1  + (0.158 − 0.273i)5-s + 0.0349i·7-s + 1.26i·11-s + (0.982 − 0.567i)13-s + (−0.297 + 0.515i)17-s + (−0.979 + 0.201i)19-s + (−0.647 + 0.373i)23-s + (0.449 + 0.779i)25-s + (−1.34 + 0.776i)29-s − 0.681·31-s + (0.00957 + 0.00552i)35-s − 0.315i·37-s + (−1.05 − 0.606i)41-s + (1.18 + 0.683i)43-s + (0.385 − 0.222i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.296 - 0.955i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.296 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.181656421\)
\(L(\frac12)\) \(\approx\) \(1.181656421\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.26 - 0.879i)T \)
good5 \( 1 + (-0.353 + 0.612i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 0.0924iT - 7T^{2} \)
11 \( 1 - 4.18iT - 11T^{2} \)
13 \( 1 + (-3.54 + 2.04i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.22 - 2.12i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.10 - 1.79i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (7.24 - 4.18i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.79T + 31T^{2} \)
37 \( 1 + 1.91iT - 37T^{2} \)
41 \( 1 + (6.72 + 3.88i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.76 - 4.48i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.64 + 1.52i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.58 + 0.912i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.91 - 5.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.70 + 9.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.334 + 0.579i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.93 - 3.34i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.20 - 2.09i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.60 - 9.70i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.7iT - 83T^{2} \)
89 \( 1 + (4.77 - 2.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.0 - 6.37i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.060934151759368186468037539242, −8.355199061811934048740620307893, −7.51322390683514958315265089646, −6.83995469989455779086245829863, −5.87007227278494441684256772926, −5.28828437065832334592219924944, −4.20488844340862760253131216870, −3.60249569130758911758757395864, −2.23123993630891317302216116203, −1.42563255138766681343992105967, 0.37184826008353158992925719640, 1.82300268399330068228112878094, 2.84115173891382052071940116833, 3.81938422717343972993942865828, 4.51720332961971249336830070011, 5.82287199288592920110124795203, 6.13783179280425069132590745925, 7.01603642913258848331193432740, 7.901950602499056547959086957841, 8.824568859221129797902588313987

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