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Properties

Label 2-720-16.13-c1-0-13
Degree $2$
Conductor $720$
Sign $-0.367 - 0.929i$
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  + (1.17 + 0.790i)2-s + (0.750 + 1.85i)4-s + (0.707 − 0.707i)5-s + 2.73i·7-s + (−0.584 + 2.76i)8-s + (1.38 − 0.270i)10-s + (−4.12 + 4.12i)11-s + (−1.37 − 1.37i)13-s + (−2.16 + 3.20i)14-s + (−2.87 + 2.78i)16-s + 4.94·17-s + (−0.292 − 0.292i)19-s + (1.84 + 0.779i)20-s + (−8.09 + 1.57i)22-s − 1.64i·23-s + ⋯
L(s)  = 1  + (0.829 + 0.558i)2-s + (0.375 + 0.926i)4-s + (0.316 − 0.316i)5-s + 1.03i·7-s + (−0.206 + 0.978i)8-s + (0.438 − 0.0855i)10-s + (−1.24 + 1.24i)11-s + (−0.382 − 0.382i)13-s + (−0.577 + 0.857i)14-s + (−0.718 + 0.695i)16-s + 1.20·17-s + (−0.0671 − 0.0671i)19-s + (0.411 + 0.174i)20-s + (−1.72 + 0.336i)22-s − 0.343i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.367 - 0.929i$
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: \((2,\ 720,\ (\ :1/2),\ -0.367 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29217 + 1.90107i\)
\(L(\frac12)\) \(\approx\) \(1.29217 + 1.90107i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.17 - 0.790i)T \)
3 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - 2.73iT - 7T^{2} \)
11 \( 1 + (4.12 - 4.12i)T - 11iT^{2} \)
13 \( 1 + (1.37 + 1.37i)T + 13iT^{2} \)
17 \( 1 - 4.94T + 17T^{2} \)
19 \( 1 + (0.292 + 0.292i)T + 19iT^{2} \)
23 \( 1 + 1.64iT - 23T^{2} \)
29 \( 1 + (-5.67 - 5.67i)T + 29iT^{2} \)
31 \( 1 - 3.95T + 31T^{2} \)
37 \( 1 + (-2.48 + 2.48i)T - 37iT^{2} \)
41 \( 1 + 8.40iT - 41T^{2} \)
43 \( 1 + (3.22 - 3.22i)T - 43iT^{2} \)
47 \( 1 - 5.19T + 47T^{2} \)
53 \( 1 + (7.20 - 7.20i)T - 53iT^{2} \)
59 \( 1 + (-6.41 + 6.41i)T - 59iT^{2} \)
61 \( 1 + (3.82 + 3.82i)T + 61iT^{2} \)
67 \( 1 + (-5.76 - 5.76i)T + 67iT^{2} \)
71 \( 1 + 7.92iT - 71T^{2} \)
73 \( 1 + 4.36iT - 73T^{2} \)
79 \( 1 + 5.56T + 79T^{2} \)
83 \( 1 + (-0.516 - 0.516i)T + 83iT^{2} \)
89 \( 1 + 6.42iT - 89T^{2} \)
97 \( 1 + 9.44T + 97T^{2} \)
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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

−10.64202445404030250372510596866, −9.831311292278147526861874350212, −8.751362264331449544076177281609, −7.931753376269952729973970299204, −7.17214593038675904600787511768, −6.00065123263601893873727931005, −5.23857365657383484850488275499, −4.65955321775821822087974618114, −3.05716790400065453472276133487, −2.20420725621699419442898805949, 0.928627967464885367336310361302, 2.60329237528370587159089846563, 3.44589308531054456548378056102, 4.58137093561350801251140477868, 5.56157724470619541678102963438, 6.37046462342636898343930248733, 7.41775923905902445664018595527, 8.313527550422565775209461352693, 9.921721552691098706471822484939, 10.11972832533711611075301545204

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