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Properties

Label 2-2736-76.31-c1-0-1
Degree $2$
Conductor $2736$
Sign $-0.938 - 0.345i$
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  + (0.767 + 1.32i)5-s − 2.31i·7-s + 5.49i·11-s + (−3.96 − 2.29i)13-s + (2.79 + 4.84i)17-s + (−1.09 + 4.21i)19-s + (−3.07 − 1.77i)23-s + (1.32 − 2.29i)25-s + (−7.12 − 4.11i)29-s − 6.20·31-s + (3.07 − 1.77i)35-s − 1.73i·37-s + (5.86 − 3.38i)41-s + (−9.30 + 5.37i)43-s + (1.68 + 0.973i)47-s + ⋯
L(s)  = 1  + (0.343 + 0.594i)5-s − 0.874i·7-s + 1.65i·11-s + (−1.10 − 0.635i)13-s + (0.678 + 1.17i)17-s + (−0.252 + 0.967i)19-s + (−0.641 − 0.370i)23-s + (0.264 − 0.458i)25-s + (−1.32 − 0.764i)29-s − 1.11·31-s + (0.519 − 0.300i)35-s − 0.284i·37-s + (0.916 − 0.528i)41-s + (−1.41 + 0.819i)43-s + (0.246 + 0.142i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 - 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $-0.938 - 0.345i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ -0.938 - 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6015903350\)
\(L(\frac12)\) \(\approx\) \(0.6015903350\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.09 - 4.21i)T \)
good5 \( 1 + (-0.767 - 1.32i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.31iT - 7T^{2} \)
11 \( 1 - 5.49iT - 11T^{2} \)
13 \( 1 + (3.96 + 2.29i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.79 - 4.84i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (3.07 + 1.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.12 + 4.11i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.20T + 31T^{2} \)
37 \( 1 + 1.73iT - 37T^{2} \)
41 \( 1 + (-5.86 + 3.38i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (9.30 - 5.37i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.68 - 0.973i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.16 + 4.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.76 + 8.24i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.32 - 4.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.20 - 7.27i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.83 - 13.5i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.79 - 11.7i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.11 + 12.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.94iT - 83T^{2} \)
89 \( 1 + (4.82 + 2.78i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.40 + 4.27i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.452087077043186191463632403039, −8.109004729812448443242621633933, −7.61407225831218400669528345519, −7.01095345362290485006965152638, −6.15240058568460179477787461854, −5.32738071215780568417550890158, −4.30351824093656488688678303675, −3.69936255905796024786099843475, −2.42046899022761611275146225811, −1.65278085354482718779517967181, 0.17937781500178389381821491997, 1.62307901269117198326200781937, 2.70961999755574693946773455156, 3.49605142936801604647360338515, 4.83395971644211764501666643460, 5.36233856945393505559103263615, 5.98364700120607030990934774432, 7.01786397241127450536452162651, 7.75611413467479138831538379345, 8.698690825084976491188917022835

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