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Properties

Label 1-6369-6369.1964-r1-0-0
Degree $1$
Conductor $6369$
Sign $-0.265 - 0.964i$
Analytic cond. $684.444$
Root an. cond. $684.444$
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  + (0.845 − 0.533i)2-s + (0.430 − 0.902i)4-s + (−0.926 + 0.376i)5-s + (−0.998 − 0.0523i)7-s + (−0.117 − 0.993i)8-s + (−0.582 + 0.812i)10-s + (−0.664 − 0.747i)13-s + (−0.872 + 0.488i)14-s + (−0.629 − 0.777i)16-s + (0.827 + 0.561i)17-s + (0.239 − 0.970i)19-s + (−0.0588 + 0.998i)20-s + (−0.555 + 0.831i)23-s + (0.716 − 0.697i)25-s + (−0.960 − 0.277i)26-s + ⋯
L(s)  = 1  + (0.845 − 0.533i)2-s + (0.430 − 0.902i)4-s + (−0.926 + 0.376i)5-s + (−0.998 − 0.0523i)7-s + (−0.117 − 0.993i)8-s + (−0.582 + 0.812i)10-s + (−0.664 − 0.747i)13-s + (−0.872 + 0.488i)14-s + (−0.629 − 0.777i)16-s + (0.827 + 0.561i)17-s + (0.239 − 0.970i)19-s + (−0.0588 + 0.998i)20-s + (−0.555 + 0.831i)23-s + (0.716 − 0.697i)25-s + (−0.960 − 0.277i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6369 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6369 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6369\)    =    \(3 \cdot 11 \cdot 193\)
Sign: $-0.265 - 0.964i$
Analytic conductor: \(684.444\)
Root analytic conductor: \(684.444\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6369} (1964, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6369,\ (1:\ ),\ -0.265 - 0.964i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8880625860 - 1.165547408i\)
\(L(\frac12)\) \(\approx\) \(0.8880625860 - 1.165547408i\)
\(L(1)\) \(\approx\) \(1.040101722 - 0.3985061219i\)
\(L(1)\) \(\approx\) \(1.040101722 - 0.3985061219i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
193 \( 1 \)
good2 \( 1 + (0.845 - 0.533i)T \)
5 \( 1 + (-0.926 + 0.376i)T \)
7 \( 1 + (-0.998 - 0.0523i)T \)
13 \( 1 + (-0.664 - 0.747i)T \)
17 \( 1 + (0.827 + 0.561i)T \)
19 \( 1 + (0.239 - 0.970i)T \)
23 \( 1 + (-0.555 + 0.831i)T \)
29 \( 1 + (-0.931 + 0.364i)T \)
31 \( 1 + (0.533 + 0.845i)T \)
37 \( 1 + (-0.998 + 0.0457i)T \)
41 \( 1 + (-0.424 + 0.905i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (-0.993 - 0.111i)T \)
53 \( 1 + (0.979 - 0.201i)T \)
59 \( 1 + (-0.824 + 0.566i)T \)
61 \( 1 + (0.997 + 0.0719i)T \)
67 \( 1 + (0.831 + 0.555i)T \)
71 \( 1 + (0.990 + 0.137i)T \)
73 \( 1 + (0.781 + 0.624i)T \)
79 \( 1 + (-0.921 + 0.388i)T \)
83 \( 1 + (0.816 - 0.577i)T \)
89 \( 1 + (-0.0980 + 0.995i)T \)
97 \( 1 + (0.768 - 0.639i)T \)
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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

−17.20049767647573394214728986619, −16.67283387004972556636797638448, −16.33920515142445280095538162122, −15.7054215049664972839454272055, −15.07915677394521884614849007883, −14.38188057542900436187453223237, −13.82902986813131842211656359807, −12.97526835118016061747833237109, −12.43795826142108227896833898963, −11.91038415543875515459649406471, −11.50411003139747305667346445695, −10.38302746530576908831536185899, −9.639347040220298758254022202188, −8.91353126236681476806147245233, −8.02087492748052686970657118961, −7.61612886906675825148973852926, −6.813934810899303266109108540306, −6.28835205527458808613823058922, −5.36868092473559893937277904186, −4.82589688841170517293783975798, −3.82442132206431906294153341908, −3.6454073747391215012806488058, −2.69848523035436373000082824693, −1.84715152621633207492399885261, −0.4635281475676408335554687020, 0.27016308652903306084239801099, 1.13592330227798126249858017706, 2.23676149080491063700365840681, 3.14141338719372899789501570212, 3.392469745358721847521657353182, 4.092582536528691445904872865514, 5.115017748000813331656780009439, 5.530781719156800829394548857526, 6.60092529068100299700236483627, 6.98899203274706588151017950739, 7.74235446269661264856359577621, 8.62768253665901414519881168668, 9.65636730717052771454634257396, 10.11828914046201399910543746005, 10.71072577936479544512334580961, 11.55439704039705125171987218468, 12.02926694579619440375402552877, 12.67346339117107050783957324194, 13.188836730664622444546167369384, 13.949417305995855146269182999487, 14.686876282977957942050781265751, 15.30192079512109122223103397043, 15.6842345596673477739114069559, 16.3736486901893765253349220063, 17.15048606537659628668091568091

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