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Properties

Label 1-5520-5520.2987-r0-0-0
Degree $1$
Conductor $5520$
Sign $0.251 + 0.967i$
Analytic cond. $25.6347$
Root an. cond. $25.6347$
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  + (−0.909 − 0.415i)7-s + (−0.281 + 0.959i)11-s + (−0.415 − 0.909i)13-s + (−0.540 + 0.841i)17-s + (0.540 + 0.841i)19-s + (0.540 − 0.841i)29-s + (0.654 − 0.755i)31-s + (−0.142 + 0.989i)37-s + (−0.142 − 0.989i)41-s + (−0.654 − 0.755i)43-s − i·47-s + (0.654 + 0.755i)49-s + (0.415 − 0.909i)53-s + (0.909 − 0.415i)59-s + (−0.755 − 0.654i)61-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)7-s + (−0.281 + 0.959i)11-s + (−0.415 − 0.909i)13-s + (−0.540 + 0.841i)17-s + (0.540 + 0.841i)19-s + (0.540 − 0.841i)29-s + (0.654 − 0.755i)31-s + (−0.142 + 0.989i)37-s + (−0.142 − 0.989i)41-s + (−0.654 − 0.755i)43-s − i·47-s + (0.654 + 0.755i)49-s + (0.415 − 0.909i)53-s + (0.909 − 0.415i)59-s + (−0.755 − 0.654i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.251 + 0.967i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (2987, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ 0.251 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7435726026 + 0.5752922977i\)
\(L(\frac12)\) \(\approx\) \(0.7435726026 + 0.5752922977i\)
\(L(1)\) \(\approx\) \(0.8528695723 + 0.04328541418i\)
\(L(1)\) \(\approx\) \(0.8528695723 + 0.04328541418i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (-0.909 - 0.415i)T \)
11 \( 1 + (-0.281 + 0.959i)T \)
13 \( 1 + (-0.415 - 0.909i)T \)
17 \( 1 + (-0.540 + 0.841i)T \)
19 \( 1 + (0.540 + 0.841i)T \)
29 \( 1 + (0.540 - 0.841i)T \)
31 \( 1 + (0.654 - 0.755i)T \)
37 \( 1 + (-0.142 + 0.989i)T \)
41 \( 1 + (-0.142 - 0.989i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.415 - 0.909i)T \)
59 \( 1 + (0.909 - 0.415i)T \)
61 \( 1 + (-0.755 - 0.654i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (0.959 - 0.281i)T \)
73 \( 1 + (0.540 + 0.841i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (-0.142 + 0.989i)T \)
89 \( 1 + (-0.654 - 0.755i)T \)
97 \( 1 + (-0.989 + 0.142i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.05573512820567415651515268405, −16.83291991316264945823106697525, −16.37170740519232097624039009488, −15.867759185913803206027019092782, −15.23910574268147198249799669469, −14.32063819362437231149695341413, −13.668657575648297205560758023949, −13.25906617657444781662751309517, −12.34545927146807023530558082478, −11.7664625515626245217393818232, −11.153034626419520784147866533061, −10.34216338664072935123918643456, −9.5806940820944795992569492782, −8.98599066419847674658667589937, −8.52244591650816570812889348383, −7.40806477395895576748941728424, −6.80255088357852165913229801601, −6.259640562126625923013944951188, −5.32761152453636784085258860948, −4.77234348688095378916381455221, −3.80370777603683965976483007467, −2.87959008233835827684983603038, −2.617350574624181681891336176923, −1.36775031239614028819531637010, −0.31692469646142852114949239391, 0.77208066559933129816328514747, 1.87793585190198465125119438252, 2.643979077350158603282221813026, 3.46199373304340500401920223236, 4.148158725895402938342581737852, 4.93630247153989303612938402449, 5.77994227081030036211223235849, 6.44936817584001918733464904591, 7.15613718779616134276732695720, 7.862985584982043017907901563330, 8.4561261787969539811420663411, 9.51681202087783143834105584497, 10.12164587310573706208456839133, 10.31087024613594413270108669893, 11.37662905999763242345117904497, 12.24425802194237545662540079941, 12.65534987298287589051638534575, 13.33757208259878992293958061908, 13.92743133749653817114203396472, 14.87011806957303069584374312670, 15.445307883782834411814004761105, 15.85001130294412986591393474822, 16.8974030775536173042427767157, 17.23274704669648140977556672861, 17.924537134724856276446007328052

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