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Properties

Label 1-675-675.241-r0-0-0
Degree $1$
Conductor $675$
Sign $0.861 - 0.508i$
Analytic cond. $3.13468$
Root an. cond. $3.13468$
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  + (0.0348 − 0.999i)2-s + (−0.997 − 0.0697i)4-s + (0.766 − 0.642i)7-s + (−0.104 + 0.994i)8-s + (−0.882 + 0.469i)11-s + (0.0348 + 0.999i)13-s + (−0.615 − 0.788i)14-s + (0.990 + 0.139i)16-s + (0.913 − 0.406i)17-s + (−0.104 + 0.994i)19-s + (0.438 + 0.898i)22-s + (−0.374 + 0.927i)23-s + 26-s + (−0.809 + 0.587i)28-s + (0.961 − 0.275i)29-s + ⋯
L(s)  = 1  + (0.0348 − 0.999i)2-s + (−0.997 − 0.0697i)4-s + (0.766 − 0.642i)7-s + (−0.104 + 0.994i)8-s + (−0.882 + 0.469i)11-s + (0.0348 + 0.999i)13-s + (−0.615 − 0.788i)14-s + (0.990 + 0.139i)16-s + (0.913 − 0.406i)17-s + (−0.104 + 0.994i)19-s + (0.438 + 0.898i)22-s + (−0.374 + 0.927i)23-s + 26-s + (−0.809 + 0.587i)28-s + (0.961 − 0.275i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.861 - 0.508i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.861 - 0.508i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227626096 - 0.3350729635i\)
\(L(\frac12)\) \(\approx\) \(1.227626096 - 0.3350729635i\)
\(L(1)\) \(\approx\) \(0.9618063937 - 0.3707260843i\)
\(L(1)\) \(\approx\) \(0.9618063937 - 0.3707260843i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.0348 - 0.999i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (-0.882 + 0.469i)T \)
13 \( 1 + (0.0348 + 0.999i)T \)
17 \( 1 + (0.913 - 0.406i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.374 + 0.927i)T \)
29 \( 1 + (0.961 - 0.275i)T \)
31 \( 1 + (-0.241 - 0.970i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
41 \( 1 + (0.0348 + 0.999i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.241 + 0.970i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.882 - 0.469i)T \)
61 \( 1 + (0.848 + 0.529i)T \)
67 \( 1 + (0.961 + 0.275i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (0.669 - 0.743i)T \)
79 \( 1 + (0.961 - 0.275i)T \)
83 \( 1 + (0.559 + 0.829i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (-0.719 - 0.694i)T \)
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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

−23.09615672245663851030873432299, −21.9512839563440248684245923846, −21.49815372995718870971704369846, −20.50346779946684134005791341838, −19.30822562217895509621569583124, −18.40376429178604988640391818261, −17.89896039856512108877796564294, −17.1026971652928891615665110576, −16.00860257094815768420533201859, −15.50449889272315101819490962491, −14.62038493985896765066420210750, −13.94655713239521448316366135026, −12.86442716798744665553332192901, −12.26440054840348219434211725097, −10.872491783296278262437149125931, −10.14192399056883875237008958803, −8.83730523580214222731849357060, −8.27851892363778442556874877370, −7.5518236418909428819974314935, −6.39209182607798454226678034270, −5.38188796069928137687007056118, −4.992179883589141481014086876291, −3.603788371686378862799783613983, −2.4640590825926432948284332676, −0.73592024185880834069796079697, 1.164886908562710463740688368486, 2.04702005951850918861651012432, 3.221873842997855575667563803779, 4.30766717894011364633743744530, 4.931832275548215208231654179122, 6.101175159871712645340034584868, 7.70295033712045934740560721239, 8.05235213538300398297395764576, 9.51543660732673700668402183837, 10.00840981404046592691820345514, 11.02434304141097099334866088794, 11.689349231461057180987420731729, 12.5336180487078335404330208633, 13.538120269952515722821082054154, 14.16422862545265722581174289699, 14.93747924109531777727968450748, 16.27823893995562104855674416869, 17.11269775641242913603086845005, 18.00642079456464493701774499010, 18.623040506553949249701295095394, 19.48487026019385853945371250315, 20.442282203016576438154973056737, 20.994169361261463680462254232316, 21.53638523019363233837364822989, 22.68019737402677050705017009807

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