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Properties

Label 2-720-16.13-c1-0-19
Degree $2$
Conductor $720$
Sign $0.297 + 0.954i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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  + (−1.15 − 0.811i)2-s + (0.681 + 1.88i)4-s + (−0.707 + 0.707i)5-s − 2.18i·7-s + (0.737 − 2.73i)8-s + (1.39 − 0.244i)10-s + (0.00889 − 0.00889i)11-s + (1.72 + 1.72i)13-s + (−1.77 + 2.52i)14-s + (−3.07 + 2.56i)16-s + 5.54·17-s + (−4.94 − 4.94i)19-s + (−1.81 − 0.847i)20-s + (−0.0175 + 0.00307i)22-s + 3.01i·23-s + ⋯
L(s)  = 1  + (−0.818 − 0.574i)2-s + (0.340 + 0.940i)4-s + (−0.316 + 0.316i)5-s − 0.824i·7-s + (0.260 − 0.965i)8-s + (0.440 − 0.0773i)10-s + (0.00268 − 0.00268i)11-s + (0.479 + 0.479i)13-s + (−0.473 + 0.674i)14-s + (−0.767 + 0.640i)16-s + 1.34·17-s + (−1.13 − 1.13i)19-s + (−0.405 − 0.189i)20-s + (−0.00373 + 0.000656i)22-s + 0.628i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.297 + 0.954i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.297 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.743585 - 0.546858i\)
\(L(\frac12)\) \(\approx\) \(0.743585 - 0.546858i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.811i)T \)
3 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + 2.18iT - 7T^{2} \)
11 \( 1 + (-0.00889 + 0.00889i)T - 11iT^{2} \)
13 \( 1 + (-1.72 - 1.72i)T + 13iT^{2} \)
17 \( 1 - 5.54T + 17T^{2} \)
19 \( 1 + (4.94 + 4.94i)T + 19iT^{2} \)
23 \( 1 - 3.01iT - 23T^{2} \)
29 \( 1 + (3.20 + 3.20i)T + 29iT^{2} \)
31 \( 1 - 3.58T + 31T^{2} \)
37 \( 1 + (-4.97 + 4.97i)T - 37iT^{2} \)
41 \( 1 + 3.76iT - 41T^{2} \)
43 \( 1 + (-6.81 + 6.81i)T - 43iT^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + (-0.932 + 0.932i)T - 53iT^{2} \)
59 \( 1 + (-4.60 + 4.60i)T - 59iT^{2} \)
61 \( 1 + (-4.17 - 4.17i)T + 61iT^{2} \)
67 \( 1 + (11.0 + 11.0i)T + 67iT^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 - 7.12iT - 73T^{2} \)
79 \( 1 + 3.41T + 79T^{2} \)
83 \( 1 + (5.31 + 5.31i)T + 83iT^{2} \)
89 \( 1 + 5.06iT - 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−10.38232537456528764351992186952, −9.425521396952233433807478624035, −8.644439355718622416097847472905, −7.59618994556558968953670409352, −7.14064803003372002302091643703, −5.98981090272319475023654807037, −4.30842214425890605246473955964, −3.59446115408585722591748228490, −2.30050106721583968809051761280, −0.74740466293142742329523861445, 1.21481106841268343624948487505, 2.72092529929707616940033060154, 4.29030560316755991725455244412, 5.62003442142529416038912506069, 6.03226166597349462706815380130, 7.29211876907327319008768362580, 8.238498761847433253857777002229, 8.585023161987151004270037619194, 9.662561978386954560676808449553, 10.36399414191297729747876180600

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