Properties

Label 1-4560-4560.29-r0-0-0
Degree $1$
Conductor $4560$
Sign $0.997 + 0.0667i$
Analytic cond. $21.1765$
Root an. cond. $21.1765$
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  + (−0.5 − 0.866i)7-s + (−0.866 − 0.5i)11-s + (−0.984 + 0.173i)13-s + (−0.939 + 0.342i)17-s + (−0.766 + 0.642i)23-s + (0.342 − 0.939i)29-s + (0.5 + 0.866i)31-s i·37-s + (−0.173 + 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.939 − 0.342i)47-s + (−0.5 + 0.866i)49-s + (0.642 + 0.766i)53-s + (0.342 + 0.939i)59-s + (0.642 + 0.766i)61-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)7-s + (−0.866 − 0.5i)11-s + (−0.984 + 0.173i)13-s + (−0.939 + 0.342i)17-s + (−0.766 + 0.642i)23-s + (0.342 − 0.939i)29-s + (0.5 + 0.866i)31-s i·37-s + (−0.173 + 0.984i)41-s + (−0.642 + 0.766i)43-s + (−0.939 − 0.342i)47-s + (−0.5 + 0.866i)49-s + (0.642 + 0.766i)53-s + (0.342 + 0.939i)59-s + (0.642 + 0.766i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0667i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.997 + 0.0667i$
Analytic conductor: \(21.1765\)
Root analytic conductor: \(21.1765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4560,\ (0:\ ),\ 0.997 + 0.0667i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8326848308 + 0.02782227713i\)
\(L(\frac12)\) \(\approx\) \(0.8326848308 + 0.02782227713i\)
\(L(1)\) \(\approx\) \(0.7764085238 - 0.06623581959i\)
\(L(1)\) \(\approx\) \(0.7764085238 - 0.06623581959i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (-0.766 + 0.642i)T \)
29 \( 1 + (0.342 - 0.939i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (-0.642 + 0.766i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 + (0.342 + 0.939i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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

−18.20479282279067149535653565603, −17.66679259900437720465319520973, −16.90484192399982486146276229217, −15.997824728006949732414996825, −15.63511578916255557091536485120, −14.931093438131657254630448221809, −14.312398755466831273507124052770, −13.28551937443078675640047081846, −12.88113932031966577730190341870, −12.10495979172985703561792895362, −11.640866577989831373125087835503, −10.60192504576043150291254028173, −9.970729563520172999912464683458, −9.45731505806798020342750104335, −8.50817975346482607383242172218, −8.04961524020096542860459507277, −6.96062416848686787769118866429, −6.59800432501754979524160851991, −5.49086815578762285892296961980, −5.0468424441069735742036223029, −4.23225131521449969981167654522, −3.156364968780250788746198735836, −2.42683480768324582864922915212, −1.98573054275813401262096942788, −0.393656201555766215480030305674, 0.51789686686601776016954962733, 1.70796641731431437506834387847, 2.604219035476115182947206609259, 3.306847411565384929900719294307, 4.23502717692025101463539075587, 4.78872075458456861036035567676, 5.75924290471585754896331653341, 6.465674199597017808190129247319, 7.20060860321682713753246488759, 7.856865215142225125888479768769, 8.532511761606339188867808502103, 9.51457818863270972062227738643, 10.074675876188926066093954468717, 10.650443823733066492477159904853, 11.4229994809844847169832783013, 12.15757677534667573505570153339, 12.99839968590570809718047631585, 13.49342708452970703118836304965, 14.04422147839165063109665473702, 14.95396446056747383372046854237, 15.58053042896583307145954566828, 16.34882878132309168217407728145, 16.73705189797480150404315203919, 17.78566700151978583330929950075, 17.92372158216313636391162173105

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