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Properties

Label 2-1520-95.87-c0-0-1
Degree $2$
Conductor $1520$
Sign $-0.970 + 0.240i$
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  + (−0.366 − 1.36i)3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)9-s − 11-s + (−1.36 − 0.366i)13-s + (−1.36 − 0.366i)15-s + (−1.36 + 0.366i)17-s + (0.866 − 0.5i)19-s + (−0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s + 31-s + (0.366 + 1.36i)33-s + 2i·39-s + 0.999i·45-s + (−0.366 + 1.36i)47-s + ⋯
L(s)  = 1  + (−0.366 − 1.36i)3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)9-s − 11-s + (−1.36 − 0.366i)13-s + (−1.36 − 0.366i)15-s + (−1.36 + 0.366i)17-s + (0.866 − 0.5i)19-s + (−0.499 − 0.866i)25-s + (0.866 − 0.5i)29-s + 31-s + (0.366 + 1.36i)33-s + 2i·39-s + 0.999i·45-s + (−0.366 + 1.36i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.970 + 0.240i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1520} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1520,\ (\ :0),\ -0.970 + 0.240i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7665278416\)
\(L(\frac12)\) \(\approx\) \(0.7665278416\)
\(L(1)\) 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 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
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 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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

−9.282124882029303099460616787154, −8.211563942264498562070193662109, −7.75950990872924109775809931247, −6.83489551188174970808547605139, −6.14177589626758685880238497923, −5.17178750828209285659193598393, −4.60808588127343893178585286012, −2.71593314429826386651061778757, −1.99166044640021880568417833873, −0.60317313529094921947958539653, 2.32879164395232944609042261008, 3.09337993542315888276715793440, 4.32477754149965386822497590930, 5.00576589367413294353494636568, 5.71737492137374738967681861365, 6.81208185502522603705687146079, 7.47569505802899701082399150990, 8.682333832687822635648852699301, 9.548335871207402681794073881754, 10.15096872289216524155979927584

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