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Properties

Label 1-5520-5520.77-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.554 - 0.832i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.554 - 0.832i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.554 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8323004867 - 1.554689152i\)
\(L(\frac12)\) \(\approx\) \(0.8323004867 - 1.554689152i\)
\(L(1)\) \(\approx\) \(1.091611007 - 0.3791756079i\)
\(L(1)\) \(\approx\) \(1.091611007 - 0.3791756079i\)

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

−17.87921375531387285515330660002, −17.66502086193409655976220276517, −16.84005261055216302967655939049, −16.011975236461480437203344405862, −15.483156505288377025226058817988, −14.70229552769186705277459345799, −14.29988273234285594934038962689, −13.49767347774308703120364638792, −12.7411144948724116477182915835, −12.00325932146131649822960949932, −11.59270688004640898088989278193, −10.71004381836395992147229985865, −10.17087448225796691245708051020, −9.27298391222365683532630024807, −8.68528597826791586527312679306, −7.96244645305264808892915000660, −7.359935030540501118820903982950, −6.47941633134664692778008469172, −5.82916017745268369896579270423, −4.94588213313595813726180272251, −4.39645312119416837061169454841, −3.672837110367262958594596732697, −2.562146734383261514793763947097, −1.82399044809312324134171924593, −1.31374636608190535517338433078, 0.47544799159750897931102007756, 1.13296735133107811792761627684, 2.24582822110366401788209188950, 2.99049715389395695413471643345, 3.74856847041665824508404786721, 4.63229177481780170027575651453, 5.32580715249099795199885489638, 5.83785457937755370219128836497, 6.95691443647881159809244795382, 7.43074152908315967640499229921, 8.28085841807807897152606025627, 8.73678974005796221641790340874, 9.56137888429966193630949533127, 10.47229293767520068838337445975, 11.11074904696965140630117579597, 11.35062704447144005650679244746, 12.34394252006682280113357973210, 13.169767785935456399218578005468, 13.69984473708027230946625842431, 14.21880276909690593696770171350, 15.04110706082976937705263434629, 15.733369145782409596594248027490, 16.230471257059195883889403583201, 17.06267107223975431835407901413, 17.7149568815602519312525117208

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