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Properties

Label 2-2736-76.27-c1-0-34
Degree $2$
Conductor $2736$
Sign $0.798 + 0.601i$
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  + (1.19 − 2.07i)5-s − 1.22i·7-s + 2.95i·11-s + (3 − 1.73i)13-s + (−0.138 + 0.239i)17-s + (1.30 + 4.15i)19-s + (−1.05 + 0.610i)23-s + (−0.361 − 0.626i)25-s + (6.58 − 3.80i)29-s + 8.90·31-s + (−2.53 − 1.46i)35-s + 7.60i·37-s + (−8.26 − 4.77i)41-s + (3 + 1.73i)43-s + (4.05 − 2.34i)47-s + ⋯
L(s)  = 1  + (0.534 − 0.926i)5-s − 0.461i·7-s + 0.890i·11-s + (0.832 − 0.480i)13-s + (−0.0335 + 0.0581i)17-s + (0.299 + 0.954i)19-s + (−0.220 + 0.127i)23-s + (−0.0723 − 0.125i)25-s + (1.22 − 0.706i)29-s + 1.59·31-s + (−0.427 − 0.246i)35-s + 1.25i·37-s + (−1.29 − 0.744i)41-s + (0.457 + 0.264i)43-s + (0.591 − 0.341i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.798 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.798 + 0.601i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.798 + 0.601i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.201684332\)
\(L(\frac12)\) \(\approx\) \(2.201684332\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 - 4.15i)T \)
good5 \( 1 + (-1.19 + 2.07i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.22iT - 7T^{2} \)
11 \( 1 - 2.95iT - 11T^{2} \)
13 \( 1 + (-3 + 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.138 - 0.239i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.05 - 0.610i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-6.58 + 3.80i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.90T + 31T^{2} \)
37 \( 1 - 7.60iT - 37T^{2} \)
41 \( 1 + (8.26 + 4.77i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3 - 1.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.05 + 2.34i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 + 3.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.44 - 5.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.19 + 5.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.83 + 6.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.75 + 3.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.392 + 0.679i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.31iT - 83T^{2} \)
89 \( 1 + (-2.58 + 1.49i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.67 + 3.27i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.590123752322355224972509674026, −8.195174352971369966931557559774, −7.26073301592385444348760693112, −6.36464499613538340333271200052, −5.65760745583369973233133452489, −4.78652644485130292972499060511, −4.13937190866001121777817306220, −3.04006106978366341869949307552, −1.76752311230396762560426070665, −0.920157602033479632796129825025, 1.04607151737966784603796879161, 2.46931610687737661120071575638, 2.99510669085760431403876036369, 4.07095388147446524717357458446, 5.11818118421470486706249496021, 6.04067216294849933849816091061, 6.48824760737890763454536781053, 7.22000807618668889290296564844, 8.351914228398090050080430729225, 8.816838352180307019008541843617

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