Properties

Label 1-675-675.58-r1-0-0
Degree $1$
Conductor $675$
Sign $-0.839 - 0.543i$
Analytic cond. $72.5388$
Root an. cond. $72.5388$
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  + (−0.0697 + 0.997i)2-s + (−0.990 − 0.139i)4-s + (−0.984 − 0.173i)7-s + (0.207 − 0.978i)8-s + (0.559 − 0.829i)11-s + (−0.0697 − 0.997i)13-s + (0.241 − 0.970i)14-s + (0.961 + 0.275i)16-s + (−0.743 − 0.669i)17-s + (0.978 + 0.207i)19-s + (0.788 + 0.615i)22-s + (0.694 − 0.719i)23-s + 26-s + (0.951 + 0.309i)28-s + (−0.848 + 0.529i)29-s + ⋯
L(s)  = 1  + (−0.0697 + 0.997i)2-s + (−0.990 − 0.139i)4-s + (−0.984 − 0.173i)7-s + (0.207 − 0.978i)8-s + (0.559 − 0.829i)11-s + (−0.0697 − 0.997i)13-s + (0.241 − 0.970i)14-s + (0.961 + 0.275i)16-s + (−0.743 − 0.669i)17-s + (0.978 + 0.207i)19-s + (0.788 + 0.615i)22-s + (0.694 − 0.719i)23-s + 26-s + (0.951 + 0.309i)28-s + (−0.848 + 0.529i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(72.5388\)
Root analytic conductor: \(72.5388\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (1:\ ),\ -0.839 - 0.543i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03553373382 - 0.1202166585i\)
\(L(\frac12)\) \(\approx\) \(0.03553373382 - 0.1202166585i\)
\(L(1)\) \(\approx\) \(0.7100586382 + 0.1904871203i\)
\(L(1)\) \(\approx\) \(0.7100586382 + 0.1904871203i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.0697 + 0.997i)T \)
7 \( 1 + (-0.984 - 0.173i)T \)
11 \( 1 + (0.559 - 0.829i)T \)
13 \( 1 + (-0.0697 - 0.997i)T \)
17 \( 1 + (-0.743 - 0.669i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (0.694 - 0.719i)T \)
29 \( 1 + (-0.848 + 0.529i)T \)
31 \( 1 + (-0.882 + 0.469i)T \)
37 \( 1 + (0.994 + 0.104i)T \)
41 \( 1 + (-0.997 + 0.0697i)T \)
43 \( 1 + (-0.342 - 0.939i)T \)
47 \( 1 + (-0.469 + 0.882i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (-0.559 - 0.829i)T \)
61 \( 1 + (0.438 + 0.898i)T \)
67 \( 1 + (0.529 - 0.848i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (0.994 - 0.104i)T \)
79 \( 1 + (-0.848 + 0.529i)T \)
83 \( 1 + (-0.927 - 0.374i)T \)
89 \( 1 + (0.104 + 0.994i)T \)
97 \( 1 + (0.999 - 0.0348i)T \)
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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

−22.67348663940165161415654293967, −21.97333118308936903239642772298, −21.369043968634813840419630414700, −20.13219810825374272349857857811, −19.8195709031909648939382021192, −18.91418684180901108196860056361, −18.21824139592079813460935706235, −17.20440948859976004940083515558, −16.54816578727954326889270461576, −15.31539852611471759859519453469, −14.50418261251642401387931023156, −13.32485871637012697104725123154, −12.96641229728286620242647831189, −11.81532328564795571092243011381, −11.36490278257673072537641632355, −10.09421445729965021133748419, −9.43206646353836240193621329279, −8.90432195662187659551784037620, −7.50226854862160845890953641380, −6.55869694829539857499063162283, −5.37066714705350564014802840639, −4.2382125170171471990069702458, −3.52386795273057746077433827183, −2.35855685853612537608010458234, −1.44368483904053221607413227942, 0.03669670909341250597427351715, 0.986390193529622162660752581165, 2.99862357051031627590155849019, 3.753348867705297246000085199223, 5.04371463650976937526784403349, 5.85507801074253331990903483311, 6.754486277272783163313766188414, 7.45540837961349104128969847434, 8.611864562836381409732122368584, 9.27851837845478973892295217, 10.14750781654765220463997064940, 11.18420348332516631984302784135, 12.50455888497446494573489909315, 13.22800224303704551486384715549, 13.93888102881742011725857183904, 14.882338515023069426801206367286, 15.733539550925941087488703528132, 16.44508236075387909936184857024, 17.00691781238948179643102212491, 18.15360394306930400468010435032, 18.67200128237003958566065993507, 19.71950836765889266873202728316, 20.37655759025290015237686437634, 21.88998641820341269528773613097, 22.35716898218788505215585010337

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