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Properties

Label 2-2736-57.50-c1-0-15
Degree $2$
Conductor $2736$
Sign $0.600 - 0.799i$
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  + (1.22 − 0.707i)5-s − 1.44·7-s − 0.635i·11-s + (5.17 + 2.98i)13-s + (−1.77 + 1.02i)17-s + (−4 + 1.73i)19-s + (4.89 + 2.82i)23-s + (−1.50 + 2.59i)25-s + (−1.22 + 2.12i)29-s + 0.953i·31-s + (−1.77 + 1.02i)35-s − 2.51i·37-s + (1.94 + 3.37i)43-s + (−1.77 − 1.02i)47-s − 4.89·49-s + ⋯
L(s)  = 1  + (0.547 − 0.316i)5-s − 0.547·7-s − 0.191i·11-s + (1.43 + 0.828i)13-s + (−0.430 + 0.248i)17-s + (−0.917 + 0.397i)19-s + (1.02 + 0.589i)23-s + (−0.300 + 0.519i)25-s + (−0.227 + 0.393i)29-s + 0.171i·31-s + (−0.300 + 0.173i)35-s − 0.412i·37-s + (0.297 + 0.514i)43-s + (−0.258 − 0.149i)47-s − 0.699·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.600 - 0.799i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.758958792\)
\(L(\frac12)\) \(\approx\) \(1.758958792\)
\(L(\frac{3}{2})\) 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 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (-1.22 + 0.707i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 + 0.635iT - 11T^{2} \)
13 \( 1 + (-5.17 - 2.98i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.77 - 1.02i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-4.89 - 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.22 - 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.953iT - 31T^{2} \)
37 \( 1 + 2.51iT - 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.94 - 3.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.77 + 1.02i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.44 + 4.24i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.22 - 12.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.72 + 2.98i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.84 - 5.10i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3 - 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.05 - 1.81i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.825 - 0.476i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 11.8iT - 83T^{2} \)
89 \( 1 + (-6.12 + 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (16.3 - 9.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.882256118653171138686797366413, −8.473360425602548277300420233789, −7.31051089246755420859181399410, −6.51113791260900802181592646338, −5.98157773853484316821630077618, −5.14350651948415531947143695755, −4.05461130635036470668516310782, −3.41127217434658786234463635615, −2.12774421401473185936221484510, −1.19262419729026175686604415984, 0.61720684680204542347803852025, 2.05283034014670109612566851262, 2.95415841897493586841199976714, 3.83445524858426397345178470961, 4.81860962682089023131791456127, 5.81034345443413883564674616634, 6.42258988144769820688750776723, 6.96679340860132734359773111582, 8.142671934918184499310316478858, 8.650148823827170479490943977554

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