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Properties

Label 2-2736-76.27-c1-0-41
Degree $2$
Conductor $2736$
Sign $-0.514 + 0.857i$
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  − 5.19i·7-s + (−1.5 + 0.866i)13-s + (4 − 1.73i)19-s + (2.5 + 4.33i)25-s + 11·31-s − 12.1i·37-s + (−10.5 − 6.06i)43-s − 20·49-s + (−6.5 − 11.2i)61-s + (−2.5 − 4.33i)67-s + (−3.5 + 6.06i)73-s + (−6.5 + 11.2i)79-s + (4.5 + 7.79i)91-s + (−12 − 6.92i)97-s + 13·103-s + ⋯
L(s)  = 1  − 1.96i·7-s + (−0.416 + 0.240i)13-s + (0.917 − 0.397i)19-s + (0.5 + 0.866i)25-s + 1.97·31-s − 1.99i·37-s + (−1.60 − 0.924i)43-s − 2.85·49-s + (−0.832 − 1.44i)61-s + (−0.305 − 0.529i)67-s + (−0.409 + 0.709i)73-s + (−0.731 + 1.26i)79-s + (0.471 + 0.817i)91-s + (−1.21 − 0.703i)97-s + 1.28·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.514 + 0.857i$
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: \((2,\ 2736,\ (\ :1/2),\ -0.514 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.424232995\)
\(L(\frac12)\) \(\approx\) \(1.424232995\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 5.19iT - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 11T + 31T^{2} \)
37 \( 1 + 12.1iT - 37T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.5 + 6.06i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.5 + 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.5 - 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (12 + 6.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.477167174427032942090309621137, −7.63045220819182003893668740789, −7.11757346460937024290737439873, −6.54465883357196843156794096547, −5.31379961824628724112932444678, −4.56410261551831355912584166795, −3.80898054312251139821562709701, −2.96249654476856726277418282109, −1.51493799715985035934529109271, −0.47951646416331324907246576272, 1.42848672925635331159848120899, 2.68949603577665229247403850346, 3.05282148851518485949227202947, 4.62908986657870710216832762363, 5.15882006201026845946440081859, 6.07898805639116292969712810577, 6.54803589164231758125137293313, 7.79333817988902189197563936966, 8.387449148627503355571768811530, 8.953445377401040021498522917011

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