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Properties

Label 2-2736-19.7-c1-0-21
Degree $2$
Conductor $2736$
Sign $0.0977 - 0.995i$
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  + (2 + 3.46i)5-s + 3·7-s + 4·11-s + (−2.5 + 4.33i)13-s + (4 + 1.73i)19-s + (−2 + 3.46i)23-s + (−5.49 + 9.52i)25-s + (4 − 6.92i)29-s − 31-s + (6 + 10.3i)35-s − 5·37-s + (4 + 6.92i)41-s + (−2.5 − 4.33i)43-s + (4 − 6.92i)47-s + 2·49-s + ⋯
L(s)  = 1  + (0.894 + 1.54i)5-s + 1.13·7-s + 1.20·11-s + (−0.693 + 1.20i)13-s + (0.917 + 0.397i)19-s + (−0.417 + 0.722i)23-s + (−1.09 + 1.90i)25-s + (0.742 − 1.28i)29-s − 0.179·31-s + (1.01 + 1.75i)35-s − 0.821·37-s + (0.624 + 1.08i)41-s + (−0.381 − 0.660i)43-s + (0.583 − 1.01i)47-s + 0.285·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.0977 - 0.995i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.0977 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.663652150\)
\(L(\frac12)\) \(\approx\) \(2.663652150\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4 - 1.73i)T \)
good5 \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8 + 13.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.5 - 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.5 + 6.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.241431494376915632436297351459, −8.131730964228801101711620765473, −7.32817658501532653875286554678, −6.73556085055988780283667544223, −6.08866185893380106237319383682, −5.19549848884685461009015931338, −4.20094931029252274647204638510, −3.31222740790048070237403236388, −2.15422673641753045623852449635, −1.60887496822223174216876199303, 0.932705803029573658256122849041, 1.54929220257768350982780903308, 2.72696892334717888625116720639, 4.13608802513676338527472232413, 4.90668742092623487607894746040, 5.34167253995475635532829145136, 6.13993357742049147799532149302, 7.26598114923559937606200382408, 8.029850932224612115333896879655, 8.843196246827819270476044869076

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