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Properties

Label 2-1520-95.68-c0-0-1
Degree $2$
Conductor $1520$
Sign $0.649 + 0.760i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
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  + (1.36 − 0.366i)3-s + (0.5 − 0.866i)5-s + (0.866 − 0.5i)9-s − 11-s + (0.366 − 1.36i)13-s + (0.366 − 1.36i)15-s + (0.366 + 1.36i)17-s + (−0.866 + 0.5i)19-s + (−0.499 − 0.866i)25-s + (−0.866 + 0.5i)29-s + 31-s + (−1.36 + 0.366i)33-s − 2i·39-s − 0.999i·45-s + (1.36 + 0.366i)47-s + ⋯
L(s)  = 1  + (1.36 − 0.366i)3-s + (0.5 − 0.866i)5-s + (0.866 − 0.5i)9-s − 11-s + (0.366 − 1.36i)13-s + (0.366 − 1.36i)15-s + (0.366 + 1.36i)17-s + (−0.866 + 0.5i)19-s + (−0.499 − 0.866i)25-s + (−0.866 + 0.5i)29-s + 31-s + (−1.36 + 0.366i)33-s − 2i·39-s − 0.999i·45-s + (1.36 + 0.366i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ 0.649 + 0.760i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.758841080\)
\(L(\frac12)\) \(\approx\) \(1.758841080\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
good3 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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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

−9.283482543194393445340450213279, −8.654902179854906624831388537341, −7.973047140454598137816812293943, −7.68271838162834821679513383694, −6.13535701788530867251812646045, −5.56039626860217112340852220908, −4.37953384496320929350823589553, −3.34472656437383202488148688272, −2.43760425745123803217569587648, −1.42079959674970058655715566334, 2.13603182708977208928675856346, 2.65003803058191205570135173193, 3.61646106364436108482116140579, 4.55548388247017443941348163194, 5.67960615427670690167892542069, 6.77953910885854413878098939246, 7.37904385597779784827653248064, 8.311316493137022051078943885436, 9.030640140565269256894423889445, 9.697287822707312143759164646825

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