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Properties

Label 2-2736-76.27-c1-0-19
Degree $2$
Conductor $2736$
Sign $0.906 - 0.422i$
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  + (−0.337 + 0.584i)5-s − 3.59i·7-s + 1.85i·11-s + (3 − 1.73i)13-s + (−2.77 + 4.80i)17-s + (−2.83 + 3.30i)19-s + (−3.10 + 1.79i)23-s + (2.27 + 3.93i)25-s + (1.98 − 1.14i)29-s + 5.56·31-s + (2.09 + 1.21i)35-s + 2.29i·37-s + (8.84 + 5.10i)41-s + (−3 − 1.73i)43-s + (0.109 − 0.0633i)47-s + ⋯
L(s)  = 1  + (−0.150 + 0.261i)5-s − 1.35i·7-s + 0.560i·11-s + (0.832 − 0.480i)13-s + (−0.672 + 1.16i)17-s + (−0.650 + 0.759i)19-s + (−0.648 + 0.374i)23-s + (0.454 + 0.787i)25-s + (0.369 − 0.213i)29-s + 1.00·31-s + (0.354 + 0.204i)35-s + 0.377i·37-s + (1.38 + 0.797i)41-s + (−0.457 − 0.264i)43-s + (0.0160 − 0.00924i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.906 - 0.422i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.660111641\)
\(L(\frac12)\) \(\approx\) \(1.660111641\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.83 - 3.30i)T \)
good5 \( 1 + (0.337 - 0.584i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.59iT - 7T^{2} \)
11 \( 1 - 1.85iT - 11T^{2} \)
13 \( 1 + (-3 + 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.77 - 4.80i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.10 - 1.79i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.98 + 1.14i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.56T + 31T^{2} \)
37 \( 1 - 2.29iT - 37T^{2} \)
41 \( 1 + (-8.84 - 5.10i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3 + 1.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.109 + 0.0633i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6 + 3.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.60 + 13.1i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.66 + 2.87i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.93 - 8.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.94 - 6.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.67 + 4.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.61iT - 83T^{2} \)
89 \( 1 + (5.31 - 3.06i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.52 - 2.03i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.678254725824184778130621254106, −8.091120579063463079401808452940, −7.40777602386713611069150321233, −6.56081697847578555672353922593, −6.03326436615563179494570960681, −4.79141588756739202240122389060, −3.99199358104512779775218746484, −3.48164638704211024802711859301, −2.07031885534292481799669935891, −0.974512123645667544407378830245, 0.67627661382497510517764685539, 2.25282757288941192961680576392, 2.82366854276067959334999085121, 4.13591851649453379704390425620, 4.82181636686380810020879904905, 5.80121576818777801804118310918, 6.34783741260868991482428274460, 7.18626288593089888416756415335, 8.367186902189982255729153155147, 8.744742824778793305949901624944

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