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Properties

Label 2-1520-95.87-c0-0-0
Degree $2$
Conductor $1520$
Sign $-0.240 - 0.970i$
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 + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)9-s − 11-s + (−0.366 + 1.36i)15-s + (−0.866 + 0.5i)19-s + (1.36 + 0.366i)23-s + (0.499 + 0.866i)25-s + (−0.866 + 0.5i)29-s + 31-s + (−0.366 − 1.36i)33-s + (−1 − i)37-s + (−0.366 − 1.36i)43-s − 0.999·45-s − i·49-s + ⋯
L(s)  = 1  + (0.366 + 1.36i)3-s + (0.866 + 0.5i)5-s + (−0.866 + 0.5i)9-s − 11-s + (−0.366 + 1.36i)15-s + (−0.866 + 0.5i)19-s + (1.36 + 0.366i)23-s + (0.499 + 0.866i)25-s + (−0.866 + 0.5i)29-s + 31-s + (−0.366 − 1.36i)33-s + (−1 − i)37-s + (−0.366 − 1.36i)43-s − 0.999·45-s − i·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1520\)    =    \(2^{4} \cdot 5 \cdot 19\)
Sign: $-0.240 - 0.970i$
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: \((2,\ 1520,\ (\ :0),\ -0.240 - 0.970i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.346427845\)
\(L(\frac12)\) \(\approx\) \(1.346427845\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
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 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1 + i)T - iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−9.983778001765370596527938372265, −9.144545850615377531647658857029, −8.665029902997503511962824071697, −7.50874499647984755227300585428, −6.61764656840542214155740467876, −5.47171915229018466551113127337, −5.04548296768940630160904005410, −3.85742714825797744221289102037, −3.06734141660751013950374101336, −2.06693403565363103620702997277, 1.09665833210504436921779824667, 2.22237414360708887419474358030, 2.87946253115811957608783181863, 4.58151221098016506587575664451, 5.39954170656920723485993551003, 6.39551835448053721871974069214, 6.92381000980386291898951695123, 7.939747937731542504705336013551, 8.459378240193719949068270114546, 9.257717762109788659241179920520

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