Properties

Label 1-552-552.443-r0-0-0
Degree $1$
Conductor $552$
Sign $0.854 - 0.519i$
Analytic cond. $2.56347$
Root an. cond. $2.56347$
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  + (0.415 − 0.909i)5-s + (0.959 + 0.281i)7-s + (0.654 − 0.755i)11-s + (0.959 − 0.281i)13-s + (0.142 + 0.989i)17-s + (−0.142 + 0.989i)19-s + (−0.654 − 0.755i)25-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.654 − 0.755i)35-s + (−0.415 − 0.909i)37-s + (−0.415 + 0.909i)41-s + (0.841 + 0.540i)43-s + 47-s + (0.841 + 0.540i)49-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)5-s + (0.959 + 0.281i)7-s + (0.654 − 0.755i)11-s + (0.959 − 0.281i)13-s + (0.142 + 0.989i)17-s + (−0.142 + 0.989i)19-s + (−0.654 − 0.755i)25-s + (−0.142 − 0.989i)29-s + (−0.841 + 0.540i)31-s + (0.654 − 0.755i)35-s + (−0.415 − 0.909i)37-s + (−0.415 + 0.909i)41-s + (0.841 + 0.540i)43-s + 47-s + (0.841 + 0.540i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.854 - 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.854 - 0.519i$
Analytic conductor: \(2.56347\)
Root analytic conductor: \(2.56347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 552,\ (0:\ ),\ 0.854 - 0.519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.717676364 - 0.4810622651i\)
\(L(\frac12)\) \(\approx\) \(1.717676364 - 0.4810622651i\)
\(L(1)\) \(\approx\) \(1.323723351 - 0.2014824431i\)
\(L(1)\) \(\approx\) \(1.323723351 - 0.2014824431i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (0.959 + 0.281i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (0.142 + 0.989i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + (-0.415 - 0.909i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.959 - 0.281i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (-0.841 + 0.540i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.142 + 0.989i)T \)
79 \( 1 + (0.959 - 0.281i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (-0.841 - 0.540i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−23.47153979774265838251007712021, −22.44607976520431977632676111094, −21.92830971777659827517676160390, −20.78476575276378762577338803281, −20.33663561933209868152616250008, −19.09290352408228196092267595322, −18.26391985553043844614728408510, −17.68304827508435416120114936265, −16.88252172356825345759733401151, −15.65400525325890384954354094874, −14.84895312325369708009144732845, −14.07397007352407979000150885578, −13.48335047249944645000155449411, −12.12483559781571356950868898138, −11.215182483944467512071225966263, −10.67927038047112314097609151728, −9.5184433366522022742875995683, −8.71651758093872095493149193952, −7.34088939186674077648557753171, −6.90027502141899231257596298763, −5.68853796708269193999942935253, −4.629224010455184575926359261149, −3.589844398727530827460536457392, −2.34753790674209123322085967825, −1.36047360435205536112666173210, 1.16895673988751515379803967857, 1.90059093901813691561941181585, 3.57099166123791107955132468005, 4.45387285320879586961935467625, 5.702205343329669563052036499433, 6.08265547218822597782089190453, 7.79320087322742612658484855291, 8.49809250267072551699342014762, 9.131235287756631039213042697756, 10.38886140957043521870350720569, 11.251215696828770883594972674865, 12.17328784376747755443133140088, 13.00641858801627909065642427495, 13.96139726633168579744592039634, 14.66067529672099644531867397089, 15.7784921220706676528518925607, 16.63466690870805955711744006838, 17.337239919066932040340119744357, 18.15478862607605696488323482011, 19.098321685591265647542448006693, 20.04878199762102525790314669037, 21.054702968384355095777941688140, 21.271615828086517981302197685268, 22.35148177415667613229004466066, 23.50997110951647686514577846215

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