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Properties

Label 2-2736-19.11-c1-0-3
Degree $2$
Conductor $2736$
Sign $-0.996 - 0.0791i$
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  + (−1.22 + 2.12i)5-s − 1.44·7-s + 2.44·11-s + (0.5 + 0.866i)13-s + (−2.44 + 4.24i)17-s + (−1 − 4.24i)19-s + (1.77 + 3.07i)23-s + (−0.499 − 0.866i)25-s + (2.44 + 4.24i)29-s + 4.55·31-s + (1.77 − 3.07i)35-s − 10.7·37-s + (0.724 − 1.25i)43-s + (−5.44 − 9.43i)47-s − 4.89·49-s + ⋯
L(s)  = 1  + (−0.547 + 0.948i)5-s − 0.547·7-s + 0.738·11-s + (0.138 + 0.240i)13-s + (−0.594 + 1.02i)17-s + (−0.229 − 0.973i)19-s + (0.370 + 0.641i)23-s + (−0.0999 − 0.173i)25-s + (0.454 + 0.787i)29-s + 0.817·31-s + (0.300 − 0.519i)35-s − 1.77·37-s + (0.110 − 0.191i)43-s + (−0.794 − 1.37i)47-s − 0.699·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.996 - 0.0791i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.996 - 0.0791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5874198402\)
\(L(\frac12)\) \(\approx\) \(0.5874198402\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 4.24i)T \)
good5 \( 1 + (1.22 - 2.12i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.44 - 4.24i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.77 - 3.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.44 - 4.24i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.55T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.724 + 1.25i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.44 + 9.43i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.22 - 2.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.67 - 6.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.94 - 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.72 + 8.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.05 + 1.81i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.17 - 2.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.10T + 83T^{2} \)
89 \( 1 + (4.22 + 7.31i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.34 + 16.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.962187466432099286723801364523, −8.650961515673195944965709890805, −7.49790736000988744964010411802, −6.71960422957810144617178051815, −6.54941952457877933158978825289, −5.35469010939283095817886121049, −4.28509820363736693292286004017, −3.53613670243333316635878370147, −2.82366139953971022098758605595, −1.54040320242911875023823481627, 0.19829702730117330490618636113, 1.35220244792899814900131375031, 2.72738561774683117188265542442, 3.72814264786763131287996927548, 4.50862223496521088428999025979, 5.18162452601709272266897120162, 6.32622890429612200343865429299, 6.78219956560773346286000750266, 7.922100730204771977699294606786, 8.419658810204123708047517322793

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