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Properties

Label 1-5328-5328.4045-r0-0-0
Degree $1$
Conductor $5328$
Sign $0.362 + 0.931i$
Analytic cond. $24.7431$
Root an. cond. $24.7431$
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)5-s + (−0.173 − 0.984i)7-s + (−0.866 + 0.5i)11-s + (−0.984 + 0.173i)13-s + (−0.939 + 0.342i)17-s + (−0.984 + 0.173i)19-s + (0.5 − 0.866i)23-s + (−0.766 − 0.642i)25-s + (0.866 − 0.5i)29-s + (−0.5 − 0.866i)31-s + (0.984 + 0.173i)35-s + (−0.173 − 0.984i)41-s + (0.866 + 0.5i)43-s + 47-s + (−0.939 + 0.342i)49-s + ⋯
L(s)  = 1  + (−0.342 + 0.939i)5-s + (−0.173 − 0.984i)7-s + (−0.866 + 0.5i)11-s + (−0.984 + 0.173i)13-s + (−0.939 + 0.342i)17-s + (−0.984 + 0.173i)19-s + (0.5 − 0.866i)23-s + (−0.766 − 0.642i)25-s + (0.866 − 0.5i)29-s + (−0.5 − 0.866i)31-s + (0.984 + 0.173i)35-s + (−0.173 − 0.984i)41-s + (0.866 + 0.5i)43-s + 47-s + (−0.939 + 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5328\)    =    \(2^{4} \cdot 3^{2} \cdot 37\)
Sign: $0.362 + 0.931i$
Analytic conductor: \(24.7431\)
Root analytic conductor: \(24.7431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5328} (4045, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5328,\ (0:\ ),\ 0.362 + 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5714974331 + 0.3907043180i\)
\(L(\frac12)\) \(\approx\) \(0.5714974331 + 0.3907043180i\)
\(L(1)\) \(\approx\) \(0.7459048934 + 0.05600970781i\)
\(L(1)\) \(\approx\) \(0.7459048934 + 0.05600970781i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 \)
good5 \( 1 + (-0.342 + 0.939i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + T \)
53 \( 1 + (0.642 - 0.766i)T \)
59 \( 1 + (-0.984 - 0.173i)T \)
61 \( 1 + (0.642 + 0.766i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 - T \)
79 \( 1 + (0.173 + 0.984i)T \)
83 \( 1 + (0.984 + 0.173i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + T \)
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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

−17.64369979309547145418284804521, −17.27471444587269024610524227995, −16.29785608739476768751380589006, −15.84454256592479358471677779463, −15.318305422647837517850791608615, −14.66516437780758330570199524494, −13.64545051873152810224401405454, −13.04617037024082864156639235006, −12.4871902588746298730427521899, −11.967448614019384732778644902297, −11.16584892339910928787557398025, −10.49051726513322873775733945844, −9.55671225872508437894228191323, −8.88817449056836178722783662412, −8.54011379825054901166934551281, −7.69168414573376457022757380739, −6.9736738625735600928458034641, −6.01881380008580464829687948459, −5.26220512373400033198146029370, −4.88515668383365919364546553977, −4.032025213350358369452899483186, −2.9180981368055737375187443279, −2.45663264106775460796316265860, −1.45491454649130667748370666169, −0.2844610461275196274581932604, 0.58754975220545278369372654305, 2.1969820346351017350397254468, 2.40903309305183183189633569612, 3.46842632867580754816502025968, 4.33329942804839222368734756273, 4.63959062721362738537434309101, 5.874788350577972767159823597862, 6.63939767320053494499952805715, 7.16445564809919635177594156178, 7.713779858509208195017062290312, 8.46868531797855320917223002529, 9.455507642777090357515585928928, 10.263151039638699496736875637929, 10.605151280629222625899566153815, 11.15430771468873738147136618902, 12.1365881529721822216726270574, 12.75574782783678400921619486370, 13.4334149457201286972896776093, 14.15247288037679184378889105037, 14.83167642010395684257226085515, 15.29005905749486224587525766740, 16.0022732237827310293538027685, 16.88851695131350432806945722112, 17.365617267657196266427749415695, 18.02590841320266079424066966432

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