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Properties

Label 2-2736-19.7-c1-0-2
Degree $2$
Conductor $2736$
Sign $-0.514 - 0.857i$
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  + (−1.33 − 2.31i)5-s + 3.67·7-s − 3.81·11-s + (−0.0719 + 0.124i)13-s + (−4.24 + 0.990i)19-s + (−3.76 + 6.52i)23-s + (−1.07 + 1.85i)25-s + (−2.67 + 4.62i)29-s − 8.81·31-s + (−4.90 − 8.50i)35-s − 37-s + (2.67 + 4.62i)41-s + (1.40 + 2.43i)43-s + (3 − 5.19i)47-s + 6.48·49-s + ⋯
L(s)  = 1  + (−0.597 − 1.03i)5-s + 1.38·7-s − 1.15·11-s + (−0.0199 + 0.0345i)13-s + (−0.973 + 0.227i)19-s + (−0.784 + 1.35i)23-s + (−0.214 + 0.371i)25-s + (−0.496 + 0.859i)29-s − 1.58·31-s + (−0.829 − 1.43i)35-s − 0.164·37-s + (0.417 + 0.723i)41-s + (0.214 + 0.372i)43-s + (0.437 − 0.757i)47-s + 0.927·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.514 - 0.857i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.514 - 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4271802616\)
\(L(\frac12)\) \(\approx\) \(0.4271802616\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.24 - 0.990i)T \)
good5 \( 1 + (1.33 + 2.31i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3.67T + 7T^{2} \)
11 \( 1 + 3.81T + 11T^{2} \)
13 \( 1 + (0.0719 - 0.124i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.76 - 6.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.67 - 4.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.81T + 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-2.67 - 4.62i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.40 - 2.43i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.00 + 6.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.90 + 3.30i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.74 - 9.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.69 + 4.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.81 - 11.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.172 - 0.299i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.26 - 5.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.28T + 83T^{2} \)
89 \( 1 + (4.33 - 7.51i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.95 - 5.12i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.862480789071500375663717804169, −8.187830315789574586462029328525, −7.85140764172723421941555385160, −7.05196186708430595858614315235, −5.60029414157600965797959687905, −5.25192430922517229865540982157, −4.41821572399686948429587212679, −3.69518418115996276483139552655, −2.23247227691159418958044663409, −1.36224134516743826028820053926, 0.13348308783236207734273688825, 1.98475022074641266274034760751, 2.64267100416645858283394770764, 3.84870057796152887253456632305, 4.55392819542778961398097000770, 5.44387291544314930655079073398, 6.29985850475012346546616406572, 7.30939379978945440039610217872, 7.75419843325309742073839819118, 8.350291461765760633354186033005

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