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Properties

Label 2-728-104.101-c1-0-17
Degree $2$
Conductor $728$
Sign $0.331 - 0.943i$
Analytic cond. $5.81310$
Root an. cond. $2.41103$
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  + (1.28 − 0.586i)2-s + (−1.27 + 0.733i)3-s + (1.31 − 1.50i)4-s − 3.32·5-s + (−1.20 + 1.69i)6-s + (0.866 + 0.5i)7-s + (0.802 − 2.71i)8-s + (−0.422 + 0.731i)9-s + (−4.28 + 1.95i)10-s + (1.07 + 1.85i)11-s + (−0.559 + 2.88i)12-s + (3.60 + 0.134i)13-s + (1.40 + 0.135i)14-s + (4.22 − 2.44i)15-s + (−0.559 − 3.96i)16-s + (−3.58 + 6.21i)17-s + ⋯
L(s)  = 1  + (0.909 − 0.414i)2-s + (−0.733 + 0.423i)3-s + (0.655 − 0.754i)4-s − 1.48·5-s + (−0.492 + 0.690i)6-s + (0.327 + 0.188i)7-s + (0.283 − 0.958i)8-s + (−0.140 + 0.243i)9-s + (−1.35 + 0.617i)10-s + (0.323 + 0.560i)11-s + (−0.161 + 0.832i)12-s + (0.999 + 0.0373i)13-s + (0.376 + 0.0361i)14-s + (1.09 − 0.630i)15-s + (−0.139 − 0.990i)16-s + (−0.870 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(728\)    =    \(2^{3} \cdot 7 \cdot 13\)
Sign: $0.331 - 0.943i$
Analytic conductor: \(5.81310\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{728} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 728,\ (\ :1/2),\ 0.331 - 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05060 + 0.744583i\)
\(L(\frac12)\) \(\approx\) \(1.05060 + 0.744583i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.28 + 0.586i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-3.60 - 0.134i)T \)
good3 \( 1 + (1.27 - 0.733i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 3.32T + 5T^{2} \)
11 \( 1 + (-1.07 - 1.85i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.58 - 6.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.04 - 3.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.38 - 4.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.389 + 0.224i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.04iT - 31T^{2} \)
37 \( 1 + (-4.63 - 8.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.755 - 0.435i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.05 + 2.92i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.30iT - 47T^{2} \)
53 \( 1 - 4.80iT - 53T^{2} \)
59 \( 1 + (-1.24 + 2.15i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.96 - 2.86i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.25 + 2.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.6 + 6.13i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 - 2.66T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + (-2.59 + 1.50i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.7 - 7.95i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.74769103515134089053651530167, −10.31390307858729085051955664719, −8.740079751165290629943306411481, −7.964358964180579441321704256888, −6.78564263235601808881920601152, −5.95767484228595625579579052101, −4.86713480855873025096213452008, −4.17924746780478382125105463854, −3.41965146622502681691655052277, −1.63657604797726798634092817332, 0.55849920426558469770082474994, 2.83329765534844515443700943616, 3.97020590353636402596427868849, 4.65645647420823848782347481014, 5.82797663481108475467050859400, 6.72083315831160167177971790953, 7.29002273585838408700105625737, 8.290858586980911917937081744281, 8.994309924716555586126502419446, 10.98585469436492376215004648906

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