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Properties

Label 1-285-285.47-r0-0-0
Degree $1$
Conductor $285$
Sign $0.459 + 0.888i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (0.866 − 0.5i)7-s + (−0.866 − 0.5i)8-s + (0.5 − 0.866i)11-s + (0.984 − 0.173i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.342 + 0.939i)17-s + (0.984 + 0.173i)22-s + (0.642 + 0.766i)23-s + (0.5 + 0.866i)26-s + (−0.342 + 0.939i)28-s + (−0.939 − 0.342i)29-s + (−0.5 − 0.866i)31-s + (0.984 − 0.173i)32-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (0.866 − 0.5i)7-s + (−0.866 − 0.5i)8-s + (0.5 − 0.866i)11-s + (0.984 − 0.173i)13-s + (0.766 + 0.642i)14-s + (0.173 − 0.984i)16-s + (0.342 + 0.939i)17-s + (0.984 + 0.173i)22-s + (0.642 + 0.766i)23-s + (0.5 + 0.866i)26-s + (−0.342 + 0.939i)28-s + (−0.939 − 0.342i)29-s + (−0.5 − 0.866i)31-s + (0.984 − 0.173i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.459 + 0.888i$
Analytic conductor: \(1.32353\)
Root analytic conductor: \(1.32353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 285,\ (0:\ ),\ 0.459 + 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.317983621 + 0.8017100778i\)
\(L(\frac12)\) \(\approx\) \(1.317983621 + 0.8017100778i\)
\(L(1)\) \(\approx\) \(1.185696339 + 0.5462174706i\)
\(L(1)\) \(\approx\) \(1.185696339 + 0.5462174706i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.342 + 0.939i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
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 \( 1 + (-0.342 + 0.939i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 + (-0.939 + 0.342i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.766 - 0.642i)T \)
73 \( 1 + (-0.984 - 0.173i)T \)
79 \( 1 + (-0.173 + 0.984i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.342 - 0.939i)T \)
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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

−25.2526534813094095667092760906, −24.40575555965016276971121601065, −23.30673603674770398511773740200, −22.67278144088390441131188955628, −21.64455844102557353277556197089, −20.78133473042944154256299570245, −20.275186257541542490092798594424, −19.01967816201793297821960921097, −18.24511590826014028602366698224, −17.54978749836217228623230568358, −16.11293092821545220051299190880, −14.83195035356380655002130351095, −14.32612324746602218578155856501, −13.14949544092929845489122991987, −12.19038951330358266428413843506, −11.41060689011822737746743663811, −10.560329475409516544399034642517, −9.29695020088018848918716789893, −8.621424379061662823981632671800, −7.12336720143494159886426243933, −5.66576571242482281585118683847, −4.77175872316265333141949042260, −3.70078046830134350590537767238, −2.34203968613801349467081231538, −1.31023245026887796112669269414, 1.275644252473797689562214883401, 3.39118862017991723742071364049, 4.215186035227039931621278235302, 5.50663458960972736269859851640, 6.305915767216260674061641218645, 7.5930425618009035310443987547, 8.3160085562207779629945618575, 9.295776830074078977775884174930, 10.80182892607150757225160218147, 11.68149910077171116770842352226, 13.07812762400786723934864708661, 13.73929400303445693825203649080, 14.68458326953783863747117683904, 15.45609798129813724610517141272, 16.68800780517919001627753996576, 17.1564071976647212080688790545, 18.236716237451702789056837984884, 19.1150798456995073470587481498, 20.54667886721943199277485980418, 21.342109684899987557383800028312, 22.19761477629026587444421076352, 23.30789558921764041712297899952, 23.90407981149877378519575411743, 24.67462032759207626123571566869, 25.67347620604243934082414242728

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