↑

Properties

Label 1-285-285.29-r0-0-0
Degree $1$
Conductor $285$
Sign $0.877 - 0.479i$
Analytic cond. $1.32353$
Root an. cond. $1.32353$
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  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + (0.173 − 0.984i)22-s + (0.766 − 0.642i)23-s + (0.5 + 0.866i)26-s + (0.939 + 0.342i)28-s + (−0.939 − 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.173 − 0.984i)32-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.939 + 0.342i)17-s + (0.173 − 0.984i)22-s + (0.766 − 0.642i)23-s + (0.5 + 0.866i)26-s + (0.939 + 0.342i)28-s + (−0.939 − 0.342i)29-s + (0.5 + 0.866i)31-s + (−0.173 − 0.984i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.877 - 0.479i$
Analytic conductor: \(1.32353\)
Root analytic conductor: \(1.32353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 285,\ (0:\ ),\ 0.877 - 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.257599735 - 0.5769785840i\)
\(L(\frac12)\) \(\approx\) \(2.257599735 - 0.5769785840i\)
\(L(1)\) \(\approx\) \(1.850103954 - 0.3509186428i\)
\(L(1)\) \(\approx\) \(1.850103954 - 0.3509186428i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.939 - 0.342i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.173 + 0.984i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.766 - 0.642i)T \)
47 \( 1 + (-0.939 - 0.342i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.939 - 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.173 + 0.984i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
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

−25.36234502575147709193875145505, −24.67171141678798236811723362753, −23.72963901998108886382243160846, −22.89661760895190875819945370955, −22.30521289532110672137077655197, −21.09807919252504218494582596491, −20.31137383444351371041615376602, −19.74256854584340132054714585579, −17.96230537911872764380956349609, −17.29606027192602885777108741983, −16.36863604059820269480959344971, −15.16577143157844541015338402439, −14.685221264394211972010682463299, −13.42247167371699509749314355734, −12.93938858082379538170644601461, −11.58485827438338556155978687493, −10.92233313613418807259826337697, −9.60496440301741057942605301304, −8.074899380522520071897534317607, −7.299801718908730672612643470807, −6.324599589882862321844068938468, −5.00275642805102176481971515708, −4.24681399912281144632449259595, −3.058237998447680354681450232640, −1.60363142497940025892519475870, 1.546620415475638848157956261050, 2.630845198004592941374752682076, 3.91194853578760126733005884221, 4.92735923450595982562585229636, 6.025271143982967850576415906240, 6.842857994043553877794820568334, 8.45087337778972935027205827393, 9.36602489019284457783023754268, 10.90131546831456006026799271733, 11.45473840711091867055025246179, 12.38396477951756055382988660908, 13.44503538315877166129921369559, 14.31950435594398959003858866534, 15.12572273117493806571408973945, 16.060035308894880921016037251477, 17.077929115242763281355537700, 18.54197605673326842283223024918, 19.15001081105574329394214742162, 20.20943900003028268694414549373, 21.288366918456958213432588669084, 21.74020731724958943071140173956, 22.61495525022663124101646819389, 23.75557052274678225213929635341, 24.456821166745899890106941487, 25.06112382682754987735506444323

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