↑

Properties

Label 1-552-552.35-r0-0-0
Degree $1$
Conductor $552$
Sign $0.102 - 0.994i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.841 − 0.540i)5-s + (0.142 − 0.989i)7-s + (−0.415 − 0.909i)11-s + (0.142 + 0.989i)13-s + (0.654 − 0.755i)17-s + (−0.654 − 0.755i)19-s + (0.415 − 0.909i)25-s + (−0.654 + 0.755i)29-s + (0.959 − 0.281i)31-s + (−0.415 − 0.909i)35-s + (−0.841 − 0.540i)37-s + (−0.841 + 0.540i)41-s + (−0.959 − 0.281i)43-s + 47-s + (−0.959 − 0.281i)49-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)5-s + (0.142 − 0.989i)7-s + (−0.415 − 0.909i)11-s + (0.142 + 0.989i)13-s + (0.654 − 0.755i)17-s + (−0.654 − 0.755i)19-s + (0.415 − 0.909i)25-s + (−0.654 + 0.755i)29-s + (0.959 − 0.281i)31-s + (−0.415 − 0.909i)35-s + (−0.841 − 0.540i)37-s + (−0.841 + 0.540i)41-s + (−0.959 − 0.281i)43-s + 47-s + (−0.959 − 0.281i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.102 - 0.994i$
Analytic conductor: \(2.56347\)
Root analytic conductor: \(2.56347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 552,\ (0:\ ),\ 0.102 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104291536 - 0.9967149649i\)
\(L(\frac12)\) \(\approx\) \(1.104291536 - 0.9967149649i\)
\(L(1)\) \(\approx\) \(1.127876170 - 0.3822692940i\)
\(L(1)\) \(\approx\) \(1.127876170 - 0.3822692940i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−23.37696292122786269844422559006, −22.645004624567873107342926421524, −21.89709899051042895877404991827, −21.042859196149283463386812045165, −20.48623207571158418747424625323, −19.04688006794588369475099960001, −18.597020137624784990294236554287, −17.5993971668055059986093546687, −17.14914119048955507061656095698, −15.69103460971093170889741565935, −15.04358711648604353957446727820, −14.39220805811422047598029888438, −13.16581585625631527768763758168, −12.5569383785243425780177650243, −11.56547013785491828431449560295, −10.18438432073168464459715512847, −10.08304929650221467186988085050, −8.67851647064205225161340033179, −7.89059466224494406820047807191, −6.65636863704825089189378885684, −5.76541756938967516232047250660, −5.09770344519530138144954265962, −3.56763501613783080370119573010, −2.4754907612026572263705304019, −1.68423978056233755738111489724, 0.782840797246873559622212175816, 1.933199210939221111204102774662, 3.22823470666476232709345193538, 4.442069968686056800480149162593, 5.28018869377369319799673432179, 6.36467177422164280190211857251, 7.25360077825601287752671309526, 8.43756309418159248422433226212, 9.21790397996294648012659154217, 10.197732536128466410752875514544, 10.98585652071849064825715029682, 11.98873475629526943720758191879, 13.19407559709754234259910821916, 13.71903986623181113208024897155, 14.354219801928552063549114669293, 15.72327732922037364556224905830, 16.724439115450751579135077262429, 16.96217999159396153497983225731, 18.12117501914523998085551014506, 18.92233840473869947855856989684, 19.93061563526080173449344696531, 20.82522577038483890506473169613, 21.32040584025792008324062358228, 22.18354280535064804070705903045, 23.41452949921476780832326381557

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