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Properties

Label 2-720-80.69-c1-0-23
Degree $2$
Conductor $720$
Sign $0.431 + 0.902i$
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  + (0.345 − 1.37i)2-s + (−1.76 − 0.946i)4-s + (−2.16 + 0.561i)5-s + 4.51·7-s + (−1.90 + 2.08i)8-s + (0.0233 + 3.16i)10-s + (3.44 + 3.44i)11-s + (−0.113 − 0.113i)13-s + (1.55 − 6.19i)14-s + (2.20 + 3.33i)16-s − 5.03i·17-s + (−0.992 + 0.992i)19-s + (4.34 + 1.05i)20-s + (5.90 − 3.53i)22-s + 8.00·23-s + ⋯
L(s)  = 1  + (0.244 − 0.969i)2-s + (−0.880 − 0.473i)4-s + (−0.967 + 0.251i)5-s + 1.70·7-s + (−0.673 + 0.738i)8-s + (0.00739 + 0.999i)10-s + (1.03 + 1.03i)11-s + (−0.0315 − 0.0315i)13-s + (0.416 − 1.65i)14-s + (0.551 + 0.833i)16-s − 1.22i·17-s + (−0.227 + 0.227i)19-s + (0.971 + 0.236i)20-s + (1.25 − 0.752i)22-s + 1.66·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.431 + 0.902i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.431 + 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39646 - 0.880268i\)
\(L(\frac12)\) \(\approx\) \(1.39646 - 0.880268i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.345 + 1.37i)T \)
3 \( 1 \)
5 \( 1 + (2.16 - 0.561i)T \)
good7 \( 1 - 4.51T + 7T^{2} \)
11 \( 1 + (-3.44 - 3.44i)T + 11iT^{2} \)
13 \( 1 + (0.113 + 0.113i)T + 13iT^{2} \)
17 \( 1 + 5.03iT - 17T^{2} \)
19 \( 1 + (0.992 - 0.992i)T - 19iT^{2} \)
23 \( 1 - 8.00T + 23T^{2} \)
29 \( 1 + (-1.01 + 1.01i)T - 29iT^{2} \)
31 \( 1 + 6.42T + 31T^{2} \)
37 \( 1 + (-1.63 + 1.63i)T - 37iT^{2} \)
41 \( 1 - 3.35iT - 41T^{2} \)
43 \( 1 + (-5.68 + 5.68i)T - 43iT^{2} \)
47 \( 1 + 9.10iT - 47T^{2} \)
53 \( 1 + (3.27 - 3.27i)T - 53iT^{2} \)
59 \( 1 + (-5.30 - 5.30i)T + 59iT^{2} \)
61 \( 1 + (5.87 - 5.87i)T - 61iT^{2} \)
67 \( 1 + (1.87 + 1.87i)T + 67iT^{2} \)
71 \( 1 + 0.635iT - 71T^{2} \)
73 \( 1 - 6.14T + 73T^{2} \)
79 \( 1 - 1.76T + 79T^{2} \)
83 \( 1 + (-6.39 - 6.39i)T + 83iT^{2} \)
89 \( 1 - 0.579iT - 89T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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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

−10.66554849354734199788997194959, −9.372891172000841700223742436947, −8.746535536494178474627982389495, −7.70198313529657470597274842487, −6.96005460131035226794867502965, −5.24472961489310638736304375847, −4.60060414336595414973287670727, −3.80800696052301197573156746760, −2.41256213551800493934693268448, −1.15157467652019143783688639979, 1.16359248393176352561136751118, 3.43691573681154293822764757864, 4.33897722659607126047255370085, 5.06646688339600200938372342459, 6.13826267711215683668316700646, 7.20125691604472752957014639382, 7.999126353317849079495471242521, 8.597971158393324624943293026887, 9.167869774305730220792382360824, 11.01735400110204539305237596497

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