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Properties

Label 2-2736-228.83-c1-0-19
Degree $2$
Conductor $2736$
Sign $0.860 - 0.510i$
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  + (2.15 − 1.24i)5-s − 0.872i·7-s + 2.41·11-s + (−1.98 + 3.43i)13-s + (−0.303 + 0.174i)17-s + (1.35 + 4.14i)19-s + (−3.88 + 6.72i)23-s + (0.608 − 1.05i)25-s + (7.96 + 4.59i)29-s + 2.43i·31-s + (−1.08 − 1.88i)35-s + 6.18·37-s + (1.31 − 0.760i)41-s + (−5.24 + 3.02i)43-s + (−4.05 + 7.01i)47-s + ⋯
L(s)  = 1  + (0.965 − 0.557i)5-s − 0.329i·7-s + 0.726·11-s + (−0.549 + 0.951i)13-s + (−0.0734 + 0.0424i)17-s + (0.311 + 0.950i)19-s + (−0.809 + 1.40i)23-s + (0.121 − 0.210i)25-s + (1.47 + 0.853i)29-s + 0.437i·31-s + (−0.183 − 0.318i)35-s + 1.01·37-s + (0.205 − 0.118i)41-s + (−0.800 + 0.462i)43-s + (−0.590 + 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 - 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2736\)    =    \(2^{4} \cdot 3^{2} \cdot 19\)
Sign: $0.860 - 0.510i$
Analytic conductor: \(21.8470\)
Root analytic conductor: \(4.67408\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2736} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2736,\ (\ :1/2),\ 0.860 - 0.510i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.181941828\)
\(L(\frac12)\) \(\approx\) \(2.181941828\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 - 4.14i)T \)
good5 \( 1 + (-2.15 + 1.24i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + 0.872iT - 7T^{2} \)
11 \( 1 - 2.41T + 11T^{2} \)
13 \( 1 + (1.98 - 3.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.303 - 0.174i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.88 - 6.72i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.96 - 4.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.43iT - 31T^{2} \)
37 \( 1 - 6.18T + 37T^{2} \)
41 \( 1 + (-1.31 + 0.760i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.24 - 3.02i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.05 - 7.01i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.41 - 1.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.55 - 2.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.53 + 11.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.04 + 2.91i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.65 + 8.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.01 + 1.75i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.8 + 6.25i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.29T + 83T^{2} \)
89 \( 1 + (-14.0 - 8.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.90 + 15.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.081997150680578330820042349905, −8.179515319711913643405720421897, −7.34978177317100341135493362091, −6.49841484990933164516008356404, −5.87102403952370546517849455655, −4.98856537905715611643556833216, −4.22942953124543769556322877585, −3.24927526669696308152891349090, −1.91530079439238720180783772168, −1.26984805482518472884650114266, 0.75553853787900577549724318649, 2.35081847998921923238979044144, 2.67338662557657750351731834811, 4.00486611813238422593403242195, 4.93142687970226371709473769081, 5.81082218476155775698716120389, 6.42941428808788618356105650286, 7.06204219917343226172529276888, 8.120515439849206810207554451729, 8.739908136604596754732458459306

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