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Properties

Label 2-720-80.69-c1-0-35
Degree $2$
Conductor $720$
Sign $-0.287 + 0.957i$
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  + (−0.550 − 1.30i)2-s + (−1.39 + 1.43i)4-s + (2.23 − 0.162i)5-s − 2.93·7-s + (2.63 + 1.02i)8-s + (−1.43 − 2.81i)10-s + (0.663 + 0.663i)11-s + (−1.12 − 1.12i)13-s + (1.61 + 3.82i)14-s + (−0.112 − 3.99i)16-s − 7.47i·17-s + (0.423 − 0.423i)19-s + (−2.87 + 3.42i)20-s + (0.499 − 1.23i)22-s + 6.17·23-s + ⋯
L(s)  = 1  + (−0.389 − 0.921i)2-s + (−0.697 + 0.716i)4-s + (0.997 − 0.0724i)5-s − 1.10·7-s + (0.931 + 0.363i)8-s + (−0.454 − 0.890i)10-s + (0.200 + 0.200i)11-s + (−0.312 − 0.312i)13-s + (0.431 + 1.02i)14-s + (−0.0281 − 0.999i)16-s − 1.81i·17-s + (0.0971 − 0.0971i)19-s + (−0.643 + 0.765i)20-s + (0.106 − 0.262i)22-s + 1.28·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.287 + 0.957i$
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: \((2,\ 720,\ (\ :1/2),\ -0.287 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.689212 - 0.926865i\)
\(L(\frac12)\) \(\approx\) \(0.689212 - 0.926865i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.550 + 1.30i)T \)
3 \( 1 \)
5 \( 1 + (-2.23 + 0.162i)T \)
good7 \( 1 + 2.93T + 7T^{2} \)
11 \( 1 + (-0.663 - 0.663i)T + 11iT^{2} \)
13 \( 1 + (1.12 + 1.12i)T + 13iT^{2} \)
17 \( 1 + 7.47iT - 17T^{2} \)
19 \( 1 + (-0.423 + 0.423i)T - 19iT^{2} \)
23 \( 1 - 6.17T + 23T^{2} \)
29 \( 1 + (-2.95 + 2.95i)T - 29iT^{2} \)
31 \( 1 - 1.82T + 31T^{2} \)
37 \( 1 + (-5.53 + 5.53i)T - 37iT^{2} \)
41 \( 1 + 12.3iT - 41T^{2} \)
43 \( 1 + (0.897 - 0.897i)T - 43iT^{2} \)
47 \( 1 - 4.12iT - 47T^{2} \)
53 \( 1 + (-0.146 + 0.146i)T - 53iT^{2} \)
59 \( 1 + (-7.72 - 7.72i)T + 59iT^{2} \)
61 \( 1 + (7.37 - 7.37i)T - 61iT^{2} \)
67 \( 1 + (8.68 + 8.68i)T + 67iT^{2} \)
71 \( 1 + 8.95iT - 71T^{2} \)
73 \( 1 + 0.174T + 73T^{2} \)
79 \( 1 - 3.06T + 79T^{2} \)
83 \( 1 + (9.18 + 9.18i)T + 83iT^{2} \)
89 \( 1 - 8.71iT - 89T^{2} \)
97 \( 1 - 10.5iT - 97T^{2} \)
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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

−10.09513320000641064927958538003, −9.246521216750105756524914698719, −9.114326508962497078778864087062, −7.55360022219825485934389448662, −6.75968916799497728397730591012, −5.52914353451880508445583063091, −4.56495734268115229499465118311, −3.10892005245501910114999676411, −2.45151708278761410642176319112, −0.75713983667029465222383368976, 1.39971812704967092837622529912, 3.08980537702861022130194838298, 4.49917067486742249711530974461, 5.62600450604548244626540364865, 6.42781434913809490537462219863, 6.82671279572359509438619687314, 8.191262058600261037550279048612, 8.937736139736989852585804786813, 9.802882978204385678257509150533, 10.18757147202952790489080834015

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