Properties

Label 2-2736-19.18-c2-0-0
Degree $2$
Conductor $2736$
Sign $-0.972 + 0.232i$
Analytic cond. $74.5506$
Root an. cond. $8.63426$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.585·5-s + 5.15·7-s − 11.8·11-s + 14.9i·13-s + 8.91·17-s + (−18.4 + 4.41i)19-s + 35.6·23-s − 24.6·25-s − 7.12i·29-s + 10.8i·31-s − 3.02·35-s − 51.3i·37-s + 33.2i·41-s − 54.2·43-s + 32.6·47-s + ⋯
L(s)  = 1  − 0.117·5-s + 0.737·7-s − 1.07·11-s + 1.14i·13-s + 0.524·17-s + (−0.972 + 0.232i)19-s + 1.55·23-s − 0.986·25-s − 0.245i·29-s + 0.348i·31-s − 0.0863·35-s − 1.38i·37-s + 0.810i·41-s − 1.26·43-s + 0.694·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.972 + 0.232i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.972 + 0.232i$
Analytic conductor: \(74.5506\)
Root analytic conductor: \(8.63426\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1),\ -0.972 + 0.232i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.02312271598\)
\(L(\frac12)\) \(\approx\) \(0.02312271598\)
\(L(2)\) 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 \( 1 + (18.4 - 4.41i)T \)
good5 \( 1 + 0.585T + 25T^{2} \)
7 \( 1 - 5.15T + 49T^{2} \)
11 \( 1 + 11.8T + 121T^{2} \)
13 \( 1 - 14.9iT - 169T^{2} \)
17 \( 1 - 8.91T + 289T^{2} \)
23 \( 1 - 35.6T + 529T^{2} \)
29 \( 1 + 7.12iT - 841T^{2} \)
31 \( 1 - 10.8iT - 961T^{2} \)
37 \( 1 + 51.3iT - 1.36e3T^{2} \)
41 \( 1 - 33.2iT - 1.68e3T^{2} \)
43 \( 1 + 54.2T + 1.84e3T^{2} \)
47 \( 1 - 32.6T + 2.20e3T^{2} \)
53 \( 1 - 91.2iT - 2.80e3T^{2} \)
59 \( 1 + 107. iT - 3.48e3T^{2} \)
61 \( 1 + 23.6T + 3.72e3T^{2} \)
67 \( 1 + 107. iT - 4.48e3T^{2} \)
71 \( 1 + 72.7iT - 5.04e3T^{2} \)
73 \( 1 + 5.25T + 5.32e3T^{2} \)
79 \( 1 + 45.5iT - 6.24e3T^{2} \)
83 \( 1 + 132.T + 6.88e3T^{2} \)
89 \( 1 - 42.1iT - 7.92e3T^{2} \)
97 \( 1 - 143. iT - 9.40e3T^{2} \)
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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

−9.041700075026749023325676367482, −8.180638707312305864150432992000, −7.67380296195995591691002227717, −6.83703298039361314344860958528, −5.96842269562587848113112917290, −5.05007805115498546339719338303, −4.48131504161109559816887180575, −3.45601633351321163415819284471, −2.35798238564336845541091607021, −1.49247768136003596822311782386, 0.00528375849895951542518361141, 1.25107358308354890092901040471, 2.47346403655978105379669260538, 3.24937164674000419034934458919, 4.38128269535087692587985287611, 5.21889527741056974692032147250, 5.68146520040900525094509594659, 6.86488219958845321285764470292, 7.57428532355368898931837809113, 8.297064635088372380581453183749

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