Properties

Label 2-2736-19.7-c1-0-37
Degree $2$
Conductor $2736$
Sign $0.813 + 0.582i$
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  + 4·7-s + 3·11-s + (−1 + 1.73i)13-s + (−3 − 5.19i)17-s + (3.5 − 2.59i)19-s + (3 − 5.19i)23-s + (2.5 − 4.33i)25-s − 2·31-s − 10·37-s + (4.5 + 7.79i)41-s + (−2 − 3.46i)43-s + 9·49-s + (3 − 5.19i)53-s + (4.5 + 7.79i)59-s + (2 − 3.46i)61-s + ⋯
L(s)  = 1  + 1.51·7-s + 0.904·11-s + (−0.277 + 0.480i)13-s + (−0.727 − 1.26i)17-s + (0.802 − 0.596i)19-s + (0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s − 0.359·31-s − 1.64·37-s + (0.702 + 1.21i)41-s + (−0.304 − 0.528i)43-s + 1.28·49-s + (0.412 − 0.713i)53-s + (0.585 + 1.01i)59-s + (0.256 − 0.443i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.813 + 0.582i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.305641188\)
\(L(\frac12)\) \(\approx\) \(2.305641188\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 + 2.59i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.5 + 14.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.809842204645031191082320694440, −8.087322983214289735111935441254, −7.01474542960158354985191035048, −6.81407231652924487367687030366, −5.44641619484999049912524165011, −4.76003577771701243787518705418, −4.25561498081208946276202095250, −2.90224816794918410657245787754, −1.96306291386956107663605922492, −0.846566516559406949925808136369, 1.28057964645246709468571462560, 1.92066422338471273666688719567, 3.38410425171896998797815321964, 4.11220154871372790066642210530, 5.14055131386955070080165044730, 5.58974977633622392681504014934, 6.73228229957901982452071099600, 7.47048054528617358571003891896, 8.130877166547134200525843133549, 8.866965424272802510991259639094

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