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Properties

Label 1-5641-5641.1369-r0-0-0
Degree $1$
Conductor $5641$
Sign $-0.895 + 0.444i$
Analytic cond. $26.1966$
Root an. cond. $26.1966$
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  + (−0.964 + 0.264i)2-s + (0.645 − 0.763i)3-s + (0.860 − 0.509i)4-s + (0.166 − 0.986i)5-s + (−0.420 + 0.907i)6-s + (0.805 − 0.593i)7-s + (−0.695 + 0.718i)8-s + (−0.166 − 0.986i)9-s + (0.100 + 0.994i)10-s + (−0.784 − 0.619i)11-s + (0.166 − 0.986i)12-s + (0.645 + 0.763i)13-s + (−0.619 + 0.784i)14-s + (−0.645 − 0.763i)15-s + (0.480 − 0.876i)16-s + (0.328 − 0.944i)17-s + ⋯
L(s)  = 1  + (−0.964 + 0.264i)2-s + (0.645 − 0.763i)3-s + (0.860 − 0.509i)4-s + (0.166 − 0.986i)5-s + (−0.420 + 0.907i)6-s + (0.805 − 0.593i)7-s + (−0.695 + 0.718i)8-s + (−0.166 − 0.986i)9-s + (0.100 + 0.994i)10-s + (−0.784 − 0.619i)11-s + (0.166 − 0.986i)12-s + (0.645 + 0.763i)13-s + (−0.619 + 0.784i)14-s + (−0.645 − 0.763i)15-s + (0.480 − 0.876i)16-s + (0.328 − 0.944i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5641 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5641 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5641\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(26.1966\)
Root analytic conductor: \(26.1966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5641} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5641,\ (0:\ ),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2323059846 - 0.9915138993i\)
\(L(\frac12)\) \(\approx\) \(-0.2323059846 - 0.9915138993i\)
\(L(1)\) \(\approx\) \(0.7130917126 - 0.4872662571i\)
\(L(1)\) \(\approx\) \(0.7130917126 - 0.4872662571i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5641 \( 1 \)
good2 \( 1 + (-0.964 + 0.264i)T \)
3 \( 1 + (0.645 - 0.763i)T \)
5 \( 1 + (0.166 - 0.986i)T \)
7 \( 1 + (0.805 - 0.593i)T \)
11 \( 1 + (-0.784 - 0.619i)T \)
13 \( 1 + (0.645 + 0.763i)T \)
17 \( 1 + (0.328 - 0.944i)T \)
19 \( 1 + (-0.100 + 0.994i)T \)
23 \( 1 + (-0.0334 - 0.999i)T \)
29 \( 1 + (-0.876 - 0.480i)T \)
31 \( 1 + (-0.999 - 0.0334i)T \)
37 \( 1 + (0.509 + 0.860i)T \)
41 \( 1 + (-0.0667 - 0.997i)T \)
43 \( 1 + (-0.264 + 0.964i)T \)
47 \( 1 + (0.997 - 0.0667i)T \)
53 \( 1 + (-0.619 - 0.784i)T \)
59 \( 1 + (-0.824 - 0.565i)T \)
61 \( 1 + (0.876 - 0.480i)T \)
67 \( 1 + (-0.972 + 0.231i)T \)
71 \( 1 + (-0.784 + 0.619i)T \)
73 \( 1 + (0.390 + 0.920i)T \)
79 \( 1 + (0.784 - 0.619i)T \)
83 \( 1 + (0.619 + 0.784i)T \)
89 \( 1 + (-0.645 - 0.763i)T \)
97 \( 1 + (-0.805 - 0.593i)T \)
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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

−18.14587322864337471563602281768, −17.78331333242736356394648752888, −17.05247987159452536999585914506, −16.14309778983169719761391170526, −15.345248879001505892297685327717, −15.17039294377158302443163825626, −14.65585247449955783911770418316, −13.56497594204467385951330643746, −12.963170339072190141407142424488, −12.02468441570875243403751699786, −11.077483544670632622868737039391, −10.76795006033666387244745267996, −10.36441348630994623130961631453, −9.32616846400606654658200866009, −9.08714608009980293311797317101, −7.98121224443330225885843457961, −7.78784251566696282245037203173, −7.03798510949723756154904679649, −5.82438918527110703870381384273, −5.42254503521036295947012789950, −4.212937325289940032810335963760, −3.413958579899517861728006101196, −2.81146823511274916133787021693, −2.11957880362107105037597087481, −1.520358352608785872109241073333, 0.3118518789072423455960430415, 1.1240758705752416133960868238, 1.72642117333943007396555503344, 2.394959281440039228702431535107, 3.45345468200781930770100948997, 4.36374464208108288751296024593, 5.3796341214502975924616971049, 5.9526937595834063182352682799, 6.83037706742741877477105809769, 7.568128215648888134579363398459, 8.15541483540902341449830108358, 8.4702291952799278924440793210, 9.256337357756286313563349344999, 9.82612262171277783130481185971, 10.77740996349716910014476663846, 11.42962928469785472569772266528, 12.06401069086077451551530462683, 12.86899568656873659989752508606, 13.62046024967253063813011360016, 14.15384043816897131935795990240, 14.73183869262835619460027504139, 15.657705702068695287763748266641, 16.48988414329144920497712625159, 16.650807800403129171085253161170, 17.55681883627985480936616222989

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