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Properties

Label 1-6069-6069.1328-r0-0-0
Degree $1$
Conductor $6069$
Sign $0.999 + 0.0218i$
Analytic cond. $28.1843$
Root an. cond. $28.1843$
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  + (−0.988 + 0.153i)2-s + (0.952 − 0.303i)4-s + (0.662 − 0.749i)5-s + (−0.895 + 0.445i)8-s + (−0.539 + 0.842i)10-s + (0.888 + 0.459i)11-s + (0.445 + 0.895i)13-s + (0.816 − 0.577i)16-s + (−0.626 + 0.779i)19-s + (0.403 − 0.914i)20-s + (−0.948 − 0.317i)22-s + (0.999 − 0.0153i)23-s + (−0.122 − 0.992i)25-s + (−0.577 − 0.816i)26-s + (0.998 − 0.0461i)29-s + ⋯
L(s)  = 1  + (−0.988 + 0.153i)2-s + (0.952 − 0.303i)4-s + (0.662 − 0.749i)5-s + (−0.895 + 0.445i)8-s + (−0.539 + 0.842i)10-s + (0.888 + 0.459i)11-s + (0.445 + 0.895i)13-s + (0.816 − 0.577i)16-s + (−0.626 + 0.779i)19-s + (0.403 − 0.914i)20-s + (−0.948 − 0.317i)22-s + (0.999 − 0.0153i)23-s + (−0.122 − 0.992i)25-s + (−0.577 − 0.816i)26-s + (0.998 − 0.0461i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0218i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6069\)    =    \(3 \cdot 7 \cdot 17^{2}\)
Sign: $0.999 + 0.0218i$
Analytic conductor: \(28.1843\)
Root analytic conductor: \(28.1843\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6069} (1328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6069,\ (0:\ ),\ 0.999 + 0.0218i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.615789943 + 0.01762408443i\)
\(L(\frac12)\) \(\approx\) \(1.615789943 + 0.01762408443i\)
\(L(1)\) \(\approx\) \(0.9298170773 + 0.001152573281i\)
\(L(1)\) \(\approx\) \(0.9298170773 + 0.001152573281i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.988 + 0.153i)T \)
5 \( 1 + (0.662 - 0.749i)T \)
11 \( 1 + (0.888 + 0.459i)T \)
13 \( 1 + (0.445 + 0.895i)T \)
19 \( 1 + (-0.626 + 0.779i)T \)
23 \( 1 + (0.999 - 0.0153i)T \)
29 \( 1 + (0.998 - 0.0461i)T \)
31 \( 1 + (0.749 - 0.662i)T \)
37 \( 1 + (0.901 + 0.431i)T \)
41 \( 1 + (0.990 - 0.138i)T \)
43 \( 1 + (-0.995 - 0.0922i)T \)
47 \( 1 + (-0.969 + 0.243i)T \)
53 \( 1 + (0.243 - 0.969i)T \)
59 \( 1 + (-0.473 - 0.881i)T \)
61 \( 1 + (0.288 + 0.957i)T \)
67 \( 1 + (-0.153 + 0.988i)T \)
71 \( 1 + (0.873 + 0.486i)T \)
73 \( 1 + (-0.937 + 0.346i)T \)
79 \( 1 + (-0.107 - 0.994i)T \)
83 \( 1 + (0.798 - 0.602i)T \)
89 \( 1 + (0.998 - 0.0615i)T \)
97 \( 1 + (-0.873 - 0.486i)T \)
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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

−17.6764121774152593114902542801, −17.266128452970737124655352948200, −16.58469027014926806910484784437, −15.81532524388623888963443189512, −15.08854961064720967135979353204, −14.65633765434106558877948333760, −13.71319783348800030057710510666, −13.11486790910378510654291855857, −12.29091070071770974868590149041, −11.515454700256798973698148673499, −10.75286597557820426510786007488, −10.63413845308469282199254293514, −9.613092373063337377265964574740, −9.16190732780896918078316563796, −8.42270451352300063019213545177, −7.769891604417375405603448817870, −6.76945457256618614497494229891, −6.50477396641247702326817674762, −5.81293896016793179381514966186, −4.81343800784462923143374037253, −3.64307625942720706221749412957, −2.97193239812633330974477929934, −2.46226744364033525796064680978, −1.39867436346903614454115373333, −0.78387430339546337191758122144, 0.80562550844540574604908976476, 1.49485648400093905344171211672, 2.059194870800704449481495557184, 2.98829898131170556254225910663, 4.134435967584560841318416879882, 4.753052034016211814861577455794, 5.76078436474698028281883672665, 6.41716090239891616052124382306, 6.79442136254162439656154347334, 7.8350560590049904567663380393, 8.57882517544107892415397572435, 8.93329042547427834811930090612, 9.831968059676925824114536954781, 9.97979709890667210572460940099, 11.09716784327953329682466320022, 11.65451177295643557112947102911, 12.31070386686549899534094057045, 13.0355828361246550151826479412, 13.842452520514813682906315242042, 14.62129587706994851312547218135, 15.05189986411100800500630038915, 16.23055957364455581987178584328, 16.35133691211996318100499807813, 17.199027589204702802185157304973, 17.491180109833125950921990882340

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