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Properties

Label 2-2736-76.31-c1-0-13
Degree $2$
Conductor $2736$
Sign $0.211 - 0.977i$
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  + (2.05 + 3.56i)5-s − i·7-s − 4.11i·11-s + (−1.5 − 0.866i)13-s + (1.50 + 2.60i)17-s + (1.73 + 4i)19-s + (0.954 + 0.551i)23-s + (−5.96 + 10.3i)25-s + (4.51 + 2.60i)29-s − 4.26·31-s + (3.56 − 2.05i)35-s + 4.26i·37-s + (5.59 − 3.23i)43-s + (5.21 + 3.01i)47-s + 6·49-s + ⋯
L(s)  = 1  + (0.920 + 1.59i)5-s − 0.377i·7-s − 1.24i·11-s + (−0.416 − 0.240i)13-s + (0.365 + 0.632i)17-s + (0.397 + 0.917i)19-s + (0.199 + 0.114i)23-s + (−1.19 + 2.06i)25-s + (0.838 + 0.484i)29-s − 0.766·31-s + (0.602 − 0.347i)35-s + 0.701i·37-s + (0.853 − 0.492i)43-s + (0.760 + 0.439i)47-s + 0.857·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.211 - 0.977i$
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: \((2,\ 2736,\ (\ :1/2),\ 0.211 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.073297885\)
\(L(\frac12)\) \(\approx\) \(2.073297885\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 - 4i)T \)
good5 \( 1 + (-2.05 - 3.56i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + iT - 7T^{2} \)
11 \( 1 + 4.11iT - 11T^{2} \)
13 \( 1 + (1.5 + 0.866i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.50 - 2.60i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.954 - 0.551i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.51 - 2.60i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.26T + 31T^{2} \)
37 \( 1 - 4.26iT - 37T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.59 + 3.23i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.21 - 3.01i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.65 - 0.954i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.954 + 1.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.96 - 3.40i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.59 - 9.69i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.51 - 7.82i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.69 - 4.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.33 - 12.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.2iT - 83T^{2} \)
89 \( 1 + (10.6 + 6.17i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.063571337069529235734275274852, −8.140739577524970069391457434787, −7.33762839873713609584395457423, −6.72100353514438899319678013022, −5.83433992087833903784050391734, −5.53618787369763826620854118988, −3.97310375333448727661534946434, −3.20963233637660817318313688824, −2.52848009147684362217780388753, −1.27363831140442521245509540185, 0.71098449844046283438305522899, 1.86933536657529692762030037453, 2.60173392896121237251102231437, 4.18438790008364785460820601916, 4.91567040224961609372565035665, 5.31862636058481470065152596339, 6.25041912378102345304709891433, 7.21448322293959663365775108487, 7.936350883776144068988850080494, 9.037674888864686040166625821111

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