Properties

Label 1-5520-5520.59-r0-0-0
Degree $1$
Conductor $5520$
Sign $-0.579 + 0.814i$
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  + (−0.841 − 0.540i)7-s + (0.989 − 0.142i)11-s + (0.540 + 0.841i)13-s + (−0.959 + 0.281i)17-s + (0.281 − 0.959i)19-s + (−0.281 − 0.959i)29-s + (−0.415 + 0.909i)31-s + (0.755 + 0.654i)37-s + (−0.654 − 0.755i)41-s + (−0.909 + 0.415i)43-s − 47-s + (0.415 + 0.909i)49-s + (−0.540 + 0.841i)53-s + (0.540 + 0.841i)59-s + (−0.909 − 0.415i)61-s + ⋯
L(s)  = 1  + (−0.841 − 0.540i)7-s + (0.989 − 0.142i)11-s + (0.540 + 0.841i)13-s + (−0.959 + 0.281i)17-s + (0.281 − 0.959i)19-s + (−0.281 − 0.959i)29-s + (−0.415 + 0.909i)31-s + (0.755 + 0.654i)37-s + (−0.654 − 0.755i)41-s + (−0.909 + 0.415i)43-s − 47-s + (0.415 + 0.909i)49-s + (−0.540 + 0.841i)53-s + (0.540 + 0.841i)59-s + (−0.909 − 0.415i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5520\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.579 + 0.814i$
Analytic conductor: \(25.6347\)
Root analytic conductor: \(25.6347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5520} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5520,\ (0:\ ),\ -0.579 + 0.814i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2613294539 + 0.5068315542i\)
\(L(\frac12)\) \(\approx\) \(0.2613294539 + 0.5068315542i\)
\(L(1)\) \(\approx\) \(0.8752703019 + 0.01403983773i\)
\(L(1)\) \(\approx\) \(0.8752703019 + 0.01403983773i\)

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 \( 1 + (-0.841 - 0.540i)T \)
11 \( 1 + (0.989 - 0.142i)T \)
13 \( 1 + (0.540 + 0.841i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (0.281 - 0.959i)T \)
29 \( 1 + (-0.281 - 0.959i)T \)
31 \( 1 + (-0.415 + 0.909i)T \)
37 \( 1 + (0.755 + 0.654i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (-0.909 + 0.415i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.540 + 0.841i)T \)
59 \( 1 + (0.540 + 0.841i)T \)
61 \( 1 + (-0.909 - 0.415i)T \)
67 \( 1 + (0.989 + 0.142i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (-0.755 - 0.654i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.654 + 0.755i)T \)
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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

−17.71696119997642033338850356385, −16.88029967951696871732222981643, −16.291620797289086611376124744327, −15.73725405318906338888897289851, −14.961516426971142475847774846739, −14.49328887829887645260417425652, −13.55551480611074141313142383127, −12.90870887467956509239712578978, −12.50615078119793301355170365756, −11.512746457633640088152643996479, −11.194373122157684543798596252028, −10.02014416438460314878565308443, −9.722804506300040641224755091405, −8.828185238160201649100457875354, −8.38183780531348025301278556154, −7.38333766655127094373755828777, −6.664420479417959987672352451394, −6.05994248388843537480238007890, −5.44624335869757409770148657448, −4.49225617425943857657942879613, −3.59599385939767009391680989927, −3.17630132876224692081330636214, −2.13237999935530967717000075634, −1.36643140584805989489247220746, −0.15534120220264378945104972786, 1.043503534763988140992758910159, 1.82408695682623842653904925548, 2.846564189233507683343485853759, 3.62297375375487221750982537615, 4.21913821446269297899226300746, 4.91399678691910316871555328564, 6.0993457840602619700210269557, 6.56989708949679871953775657034, 7.00726226499231050503165452534, 7.976262399547640612179970246960, 8.972269847146037092865383411592, 9.19255854741484799509493241578, 10.03484699871216930955983852947, 10.81399602437130959219420259342, 11.48238742538683837684368083564, 11.98263786140447416046906576371, 13.04971640704567338185749961289, 13.39715369101932535319489203357, 14.03355094557617654253794680059, 14.78287091865963967832005913697, 15.61110177341485663969069735551, 16.08583670068123778426648470532, 16.9029136988693855436386716181, 17.21883448581750013409837650199, 18.12186965458623797112292092041

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