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Properties

Label 1-5328-5328.2533-r1-0-0
Degree $1$
Conductor $5328$
Sign $-0.753 + 0.657i$
Analytic cond. $572.573$
Root an. cond. $572.573$
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  + (−0.766 + 0.642i)5-s + (0.939 + 0.342i)7-s + (−0.866 + 0.5i)11-s + (−0.939 − 0.342i)13-s + (−0.642 + 0.766i)17-s + (−0.939 − 0.342i)19-s + (0.866 + 0.5i)23-s + (0.173 − 0.984i)25-s + (0.5 + 0.866i)29-s + (−0.866 + 0.5i)31-s + (−0.939 + 0.342i)35-s + (−0.939 − 0.342i)41-s + (0.5 − 0.866i)43-s + 47-s + (0.766 + 0.642i)49-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)5-s + (0.939 + 0.342i)7-s + (−0.866 + 0.5i)11-s + (−0.939 − 0.342i)13-s + (−0.642 + 0.766i)17-s + (−0.939 − 0.342i)19-s + (0.866 + 0.5i)23-s + (0.173 − 0.984i)25-s + (0.5 + 0.866i)29-s + (−0.866 + 0.5i)31-s + (−0.939 + 0.342i)35-s + (−0.939 − 0.342i)41-s + (0.5 − 0.866i)43-s + 47-s + (0.766 + 0.642i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5328\)    =    \(2^{4} \cdot 3^{2} \cdot 37\)
Sign: $-0.753 + 0.657i$
Analytic conductor: \(572.573\)
Root analytic conductor: \(572.573\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5328} (2533, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5328,\ (1:\ ),\ -0.753 + 0.657i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4743504971 + 1.263982871i\)
\(L(\frac12)\) \(\approx\) \(0.4743504971 + 1.263982871i\)
\(L(1)\) \(\approx\) \(0.8533658492 + 0.2527953824i\)
\(L(1)\) \(\approx\) \(0.8533658492 + 0.2527953824i\)

Euler product

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

−17.38478238743333164370906348318, −16.769612511089452959444072119499, −16.36995448989357658368523502170, −15.36674473116128278729736823503, −15.05065615387094231090554104271, −14.22536461010383158402189375410, −13.47169348634413741294091806144, −12.86731748613608084976021316525, −12.12968709963745391852534727453, −11.460156486061858427390246951893, −10.956527456402690285904739491294, −10.2414081048085468349309242659, −9.27793166614936870225928864118, −8.60246095131810057729915652045, −8.03554393147165063018392931096, −7.42759778991392481746198919364, −6.76831398509042519185148090036, −5.63267894455439967699367775874, −4.901384228281182179221012436158, −4.50514024222371305421891781534, −3.75512519532427754135023354546, −2.62638417199340613140610943529, −2.03656928260631870478788030422, −0.826942651681074357881072365880, −0.2939274250436352826238888078, 0.725480299176037561517070269420, 2.07751247757377608029373305687, 2.38139278222001752025360019468, 3.40470183858108415984195020463, 4.1950702331432607341816314671, 5.0127334848518156848126563388, 5.40159641463325661934889862860, 6.67576591167426013537093581826, 7.14813408736476723028684011295, 7.83796655229938673925443104465, 8.470584823621385836340461439791, 9.0977238386802777509765448844, 10.26671390993426491439168867674, 10.7204982775706731417106086249, 11.172317531111025875320156043341, 12.143012400784844087710131561260, 12.52360788533324929285850180626, 13.316150486155357033550779897347, 14.30179041904709442161770792336, 14.840362850223932240343596791081, 15.33906836379716702812620050643, 15.70431918313544545596374916586, 16.85328971797041550908736326159, 17.48556704658335505498941420022, 17.95920630433370400557226001195

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