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Properties

Label 1-5616-5616.3397-r0-0-0
Degree $1$
Conductor $5616$
Sign $0.367 - 0.929i$
Analytic cond. $26.0805$
Root an. cond. $26.0805$
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  + (−0.642 − 0.766i)5-s + (0.173 − 0.984i)7-s + (−0.984 − 0.173i)11-s + 17-s + (0.866 − 0.5i)19-s + (−0.173 − 0.984i)23-s + (−0.173 + 0.984i)25-s + (0.342 + 0.939i)29-s + (0.939 + 0.342i)31-s + (−0.866 + 0.5i)35-s + (0.866 + 0.5i)37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.939 − 0.342i)47-s + (−0.939 − 0.342i)49-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)5-s + (0.173 − 0.984i)7-s + (−0.984 − 0.173i)11-s + 17-s + (0.866 − 0.5i)19-s + (−0.173 − 0.984i)23-s + (−0.173 + 0.984i)25-s + (0.342 + 0.939i)29-s + (0.939 + 0.342i)31-s + (−0.866 + 0.5i)35-s + (0.866 + 0.5i)37-s + (0.766 − 0.642i)41-s + (−0.342 − 0.939i)43-s + (0.939 − 0.342i)47-s + (−0.939 − 0.342i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5616 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5616\)    =    \(2^{4} \cdot 3^{3} \cdot 13\)
Sign: $0.367 - 0.929i$
Analytic conductor: \(26.0805\)
Root analytic conductor: \(26.0805\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5616} (3397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5616,\ (0:\ ),\ 0.367 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.392122015 - 0.9466522069i\)
\(L(\frac12)\) \(\approx\) \(1.392122015 - 0.9466522069i\)
\(L(1)\) \(\approx\) \(0.9901820965 - 0.3034069980i\)
\(L(1)\) \(\approx\) \(0.9901820965 - 0.3034069980i\)

Euler product

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

−18.10928405810562132741580904696, −17.53570841304506486198890281826, −16.398005728471400404627832489117, −15.91483716055309304631652389186, −15.30997492635167385823784510190, −14.81442335336486951844440536120, −14.10513475742061925875664441022, −13.38321760991524307847689426265, −12.49766599581646383764806366491, −11.92178128549856376558078693880, −11.43320484646577442604100176945, −10.70557365055001478174750006204, −9.802378458666142495509258799175, −9.52166448095461858170714873861, −8.16073290719960656849964922860, −7.95839782762371447407990296151, −7.34623742615386475532737359203, −6.25945128616449434107529493268, −5.74912512469772988621370727787, −5.00904902555434859789475342461, −4.14109693461706961834085889251, −3.21169155996769945339166735571, −2.748901559284892893053652203973, −1.953135399347196966809891434999, −0.75955937156365410719687569672, 0.72800915726658014017492421292, 1.03390558505982112079287215487, 2.35913193094521492311627554474, 3.234859510430832387491170212007, 3.901125668307034894443761179662, 4.78489765391666298181311821593, 5.12773842597397690663127634218, 6.07525609579571487710553208813, 7.13856694102837484998878434983, 7.56996731317129458985456240922, 8.22844385619855714212451042191, 8.83241259075464654892401857909, 9.778552319057654158458358785114, 10.43069064783714897757138633541, 10.968033357253593822236714689876, 11.86423792363173219408474359411, 12.36808642491319055935611748173, 13.1003621657146149309879811714, 13.72533650223296692302445349751, 14.31719567027141874674295584835, 15.17344556647168643807312738361, 15.98291042855073329288666523020, 16.25554748136333958489115821231, 17.01075638503807679216894555908, 17.57180603350126695100074747952

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