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Properties

Label 1-6369-6369.1220-r0-0-0
Degree $1$
Conductor $6369$
Sign $-0.0482 + 0.998i$
Analytic cond. $29.5775$
Root an. cond. $29.5775$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.793 + 0.608i)5-s + (−0.5 + 0.866i)7-s + (−0.707 − 0.707i)8-s + (−0.608 − 0.793i)10-s + (−0.382 − 0.923i)13-s + (0.707 − 0.707i)14-s + (0.5 + 0.866i)16-s + (−0.130 − 0.991i)17-s + (0.793 + 0.608i)19-s + (0.382 + 0.923i)20-s + (−0.707 + 0.707i)23-s + (0.258 + 0.965i)25-s + (0.130 + 0.991i)26-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.5i)4-s + (0.793 + 0.608i)5-s + (−0.5 + 0.866i)7-s + (−0.707 − 0.707i)8-s + (−0.608 − 0.793i)10-s + (−0.382 − 0.923i)13-s + (0.707 − 0.707i)14-s + (0.5 + 0.866i)16-s + (−0.130 − 0.991i)17-s + (0.793 + 0.608i)19-s + (0.382 + 0.923i)20-s + (−0.707 + 0.707i)23-s + (0.258 + 0.965i)25-s + (0.130 + 0.991i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6369 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6369 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6369\)    =    \(3 \cdot 11 \cdot 193\)
Sign: $-0.0482 + 0.998i$
Analytic conductor: \(29.5775\)
Root analytic conductor: \(29.5775\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6369} (1220, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6369,\ (0:\ ),\ -0.0482 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7138821665 + 0.7492060480i\)
\(L(\frac12)\) \(\approx\) \(0.7138821665 + 0.7492060480i\)
\(L(1)\) \(\approx\) \(0.7407418465 + 0.1349554292i\)
\(L(1)\) \(\approx\) \(0.7407418465 + 0.1349554292i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
193 \( 1 \)
good2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (0.793 + 0.608i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.382 - 0.923i)T \)
17 \( 1 + (-0.130 - 0.991i)T \)
19 \( 1 + (0.793 + 0.608i)T \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
31 \( 1 + (-0.965 - 0.258i)T \)
37 \( 1 + (0.991 + 0.130i)T \)
41 \( 1 + (0.130 + 0.991i)T \)
43 \( 1 + T \)
47 \( 1 + (0.991 + 0.130i)T \)
53 \( 1 + (0.991 + 0.130i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (0.793 - 0.608i)T \)
67 \( 1 + (0.707 - 0.707i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (-0.991 + 0.130i)T \)
79 \( 1 + (-0.793 + 0.608i)T \)
83 \( 1 + (-0.258 - 0.965i)T \)
89 \( 1 + (-0.382 + 0.923i)T \)
97 \( 1 + (-0.965 + 0.258i)T \)
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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

−17.36342384454897640670218470921, −16.87606164959447310203778596135, −16.283598709182600532279019629187, −15.887377750181508519934213268690, −14.81518232167135316227829945546, −14.26078060883916902008412070328, −13.5802281972325981267538668693, −12.89079679012171862262065300741, −12.14369993051706776787056103838, −11.38776354655626909699335528850, −10.57218786059054392075948695163, −10.073694529872887960687878546140, −9.4663296856904461568094858828, −8.92989180474886104941341146000, −8.25946293364392329316172517534, −7.24618792757744279438842982050, −6.99942834984337407837363040643, −5.92870668096318518156470405456, −5.70671677000920041442540846235, −4.47643593148291643816841036883, −3.91497444670870110787985415885, −2.61776085579714987430519959420, −2.06306169389883404734523558743, −1.20059534028084032203884518412, −0.41984721254205912218064389665, 0.899422068442095336299357607836, 1.86304631996913444924071137764, 2.58738795149117082374036127197, 3.02685252344190372111113392552, 3.80375592124729400642810926688, 5.33555612462172176240465726979, 5.69480230134044399687918527300, 6.41951583726132673215568630611, 7.32677590055169650979034671874, 7.64778460642590130076082550064, 8.70009093153229706935478280872, 9.32926142782537005480474658915, 9.83586197494526072015289240700, 10.2492917495391499062627146293, 11.25093525879444549390750108492, 11.60413714757870238429109603499, 12.59188070390674193774005730448, 12.92636206177935804088675521105, 13.90134672799125077615159835098, 14.67655282000449374717342415485, 15.26134011808160942163999859317, 16.047157748572947061784537451530, 16.42318080709753980761050290880, 17.38267795397222437404143179244, 17.91730055614174463951170220234

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