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Properties

Label 2-720-80.19-c0-0-1
Degree $2$
Conductor $720$
Sign $-0.382 + 0.923i$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (−0.707 − 0.707i)8-s − 1.00·10-s − 1.00·16-s − 1.41i·17-s + (1 − i)19-s + (−0.707 + 0.707i)20-s + 1.41i·23-s + 1.00i·25-s + 2i·31-s + (−0.707 + 0.707i)32-s + (−1.00 − 1.00i)34-s − 1.41i·38-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.707 − 0.707i)5-s + (−0.707 − 0.707i)8-s − 1.00·10-s − 1.00·16-s − 1.41i·17-s + (1 − i)19-s + (−0.707 + 0.707i)20-s + 1.41i·23-s + 1.00i·25-s + 2i·31-s + (−0.707 + 0.707i)32-s + (−1.00 − 1.00i)34-s − 1.41i·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.172592230\)
\(L(\frac12)\) \(\approx\) \(1.172592230\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + (-1 + i)T - iT^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.58622698184824029020416472264, −9.383612710503227533419508223958, −9.042360657172713694227520769634, −7.62033086590413236406189592231, −6.85744546489755928801315468096, −5.34522420386373569232417709751, −4.95770460266557193651478883978, −3.76971133479913460119449386306, −2.83457145538099995172097371699, −1.13445124268816426143868428178, 2.51040726453255872723452292787, 3.71103122058905116286250460304, 4.32768987079081907197032935784, 5.74710953651620593703347171559, 6.36693237660249894171426637406, 7.45237123227378816983632751686, 7.970036923532743868076785153349, 8.873034328356344227895818577097, 10.18655162890125860536696265526, 10.97317434809376828717867320608

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