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Properties

Label 2-2736-19.11-c1-0-17
Degree $2$
Conductor $2736$
Sign $0.761 + 0.648i$
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  + (−0.822 + 1.42i)5-s − 3.64·7-s − 4.64·11-s + (−1 − 1.73i)13-s + (−1.67 + 4.02i)19-s + (0.822 + 1.42i)23-s + (1.14 + 1.98i)25-s + (0.822 + 1.42i)29-s + 5.64·31-s + (3 − 5.19i)35-s + 0.354·37-s + (−0.145 + 0.252i)41-s + (5.64 − 9.77i)43-s + (−2.17 − 3.77i)47-s + 6.29·49-s + ⋯
L(s)  = 1  + (−0.368 + 0.637i)5-s − 1.37·7-s − 1.40·11-s + (−0.277 − 0.480i)13-s + (−0.384 + 0.923i)19-s + (0.171 + 0.297i)23-s + (0.229 + 0.396i)25-s + (0.152 + 0.264i)29-s + 1.01·31-s + (0.507 − 0.878i)35-s + 0.0582·37-s + (−0.0227 + 0.0394i)41-s + (0.860 − 1.49i)43-s + (−0.317 − 0.550i)47-s + 0.898·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 + 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.761 + 0.648i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.761 + 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7302606705\)
\(L(\frac12)\) \(\approx\) \(0.7302606705\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.67 - 4.02i)T \)
good5 \( 1 + (0.822 - 1.42i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.64T + 7T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.822 - 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.822 - 1.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.64T + 31T^{2} \)
37 \( 1 - 0.354T + 37T^{2} \)
41 \( 1 + (0.145 - 0.252i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.64 + 9.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.17 + 3.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.29 + 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.96 + 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.468 + 0.811i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.322 - 0.559i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.35 + 2.34i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.854 - 1.47i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.93T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.85 + 3.21i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.649666853939342320047454725333, −7.907077149062976917745692982981, −7.21271637406533999752660849958, −6.51841446894699716604701758692, −5.71807633932060588040453969209, −4.92137631006050160954480695273, −3.62724726668245021473998402965, −3.14192159089391962099632388997, −2.24676868777830501788122684679, −0.34794401117931944115246479489, 0.72826437299051796752591450648, 2.50785538549592776473395199656, 3.04483043459968440243080596193, 4.35755023439305227804017246180, 4.81840677818229608780173884392, 5.94978859697278165471046718646, 6.56503581686183713881248188885, 7.44992827489553152299837806286, 8.159964325639812338645758009543, 8.964009446787225654866508739537

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