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Properties

Label 2-2736-76.27-c1-0-30
Degree $2$
Conductor $2736$
Sign $0.817 + 0.575i$
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  + (1.73 − 3.01i)5-s + 3.36i·7-s − 2.43i·11-s + (−5.02 + 2.89i)13-s + (−2.67 + 4.63i)17-s + (2.84 − 3.30i)19-s + (1.30 − 0.753i)23-s + (−3.54 − 6.13i)25-s + (4.21 − 2.43i)29-s + 10.5·31-s + (10.1 + 5.85i)35-s − 8.46i·37-s + (3.41 + 1.96i)41-s + (9.02 + 5.21i)43-s + (9.63 − 5.56i)47-s + ⋯
L(s)  = 1  + (0.777 − 1.34i)5-s + 1.27i·7-s − 0.733i·11-s + (−1.39 + 0.804i)13-s + (−0.649 + 1.12i)17-s + (0.652 − 0.757i)19-s + (0.272 − 0.157i)23-s + (−0.708 − 1.22i)25-s + (0.782 − 0.451i)29-s + 1.89·31-s + (1.71 + 0.989i)35-s − 1.39i·37-s + (0.532 + 0.307i)41-s + (1.37 + 0.794i)43-s + (1.40 − 0.811i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.817 + 0.575i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.817 + 0.575i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.001278976\)
\(L(\frac12)\) \(\approx\) \(2.001278976\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.84 + 3.30i)T \)
good5 \( 1 + (-1.73 + 3.01i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.36iT - 7T^{2} \)
11 \( 1 + 2.43iT - 11T^{2} \)
13 \( 1 + (5.02 - 2.89i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.67 - 4.63i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.30 + 0.753i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.21 + 2.43i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + 8.46iT - 37T^{2} \)
41 \( 1 + (-3.41 - 1.96i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9.02 - 5.21i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-9.63 + 5.56i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.41 + 2.54i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.58 + 9.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.80 + 4.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.03 + 5.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.15 - 10.6i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.97 - 6.88i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.19 - 2.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.59iT - 83T^{2} \)
89 \( 1 + (-5.04 + 2.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.47 - 1.43i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.679448525102542483821464617818, −8.435177325299734385054057250278, −7.22321205969660207648123478347, −6.17055644808731833822554003566, −5.69492912595560796994569078618, −4.86836826043137584650939440392, −4.29749663813379610657551370792, −2.66306509840601998893040515088, −2.12579450211214774603902145998, −0.796446418370280540828191046965, 0.998638995781118854365317813225, 2.56042243139079857508363578050, 2.87667483157631391806090516580, 4.19880347202211696206728040716, 4.94313799348681862043743234970, 5.94073299570902176578008438581, 6.84236650942641317188776863720, 7.31005726795705287658953392127, 7.73001023318291069171736483108, 9.131289733643537747988866475348

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