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Properties

Label 1-3627-3627.2371-r0-0-0
Degree $1$
Conductor $3627$
Sign $0.613 + 0.789i$
Analytic cond. $16.8437$
Root an. cond. $16.8437$
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  + (0.743 − 0.669i)2-s + (0.104 − 0.994i)4-s + (−0.866 + 0.5i)5-s + (0.406 + 0.913i)7-s + (−0.587 − 0.809i)8-s + (−0.309 + 0.951i)10-s + (−0.406 − 0.913i)11-s + (0.913 + 0.406i)14-s + (−0.978 − 0.207i)16-s + (−0.809 + 0.587i)17-s + (−0.951 − 0.309i)19-s + (0.406 + 0.913i)20-s + (−0.913 − 0.406i)22-s + (0.913 + 0.406i)23-s + (0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (0.104 − 0.994i)4-s + (−0.866 + 0.5i)5-s + (0.406 + 0.913i)7-s + (−0.587 − 0.809i)8-s + (−0.309 + 0.951i)10-s + (−0.406 − 0.913i)11-s + (0.913 + 0.406i)14-s + (−0.978 − 0.207i)16-s + (−0.809 + 0.587i)17-s + (−0.951 − 0.309i)19-s + (0.406 + 0.913i)20-s + (−0.913 − 0.406i)22-s + (0.913 + 0.406i)23-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3627 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3627 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3627\)    =    \(3^{2} \cdot 13 \cdot 31\)
Sign: $0.613 + 0.789i$
Analytic conductor: \(16.8437\)
Root analytic conductor: \(16.8437\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3627} (2371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3627,\ (0:\ ),\ 0.613 + 0.789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9988810550 + 0.4888934392i\)
\(L(\frac12)\) \(\approx\) \(0.9988810550 + 0.4888934392i\)
\(L(1)\) \(\approx\) \(1.104567240 - 0.2797049323i\)
\(L(1)\) \(\approx\) \(1.104567240 - 0.2797049323i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.743 - 0.669i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.406 + 0.913i)T \)
11 \( 1 + (-0.406 - 0.913i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.951 - 0.309i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.207 + 0.978i)T \)
43 \( 1 + (0.669 + 0.743i)T \)
47 \( 1 + (0.207 - 0.978i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.743 + 0.669i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (0.587 + 0.809i)T \)
73 \( 1 + (-0.587 + 0.809i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (-0.587 + 0.809i)T \)
97 \( 1 + (-0.406 - 0.913i)T \)
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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

−18.46128624842325301924472853453, −17.58250596549060052003904201888, −17.07677203210208587227737609736, −16.456825716171134437630843329780, −15.760139592189336743493706788493, −15.08945943787912524582825979539, −14.649101385200654442963013222646, −13.726047199027475370091685786346, −13.05330267680551854297448955291, −12.55545145052072825992455665546, −11.81513243142525798979298817523, −11.02971775463090225867211213984, −10.45292141840298700136583434821, −9.13321035750969761268311774099, −8.599635190500632879009565853533, −7.72022545084766328642594908264, −7.238731212376184125993122767346, −6.72080463120291802422317007157, −5.54643947860001400546865550084, −4.67298989968029254206881298125, −4.44265209144033363665100701920, −3.658335065272745809888148444565, −2.68878838134503964165204124583, −1.657391309251186209389605090796, −0.269992715642112134142476672847, 0.976030762206755473224886944367, 2.28429131529256945383928793961, 2.61229601741439867169517637222, 3.6351491752868345971404896442, 4.20008618796423006031575580329, 5.09351938478976516341904124690, 5.82626601696517850354617634350, 6.49900287295724700140142629520, 7.37785491580734279330862988009, 8.43270757791043013250073897223, 8.83072998186925852175111773836, 9.8892650633560302580591071418, 10.83448232974202810325149068390, 11.26994870520182325939073296444, 11.59861307771529923916261614536, 12.72032743269723427027813507565, 12.9888256153555881955924673889, 13.960412585003693300137776371349, 14.7873083777835434054914573630, 15.16605090535025976917492999674, 15.69779257662475736938658281438, 16.510008812858314478459106615294, 17.651832964896832624774173291810, 18.352932261333989724191443608994, 19.02678500210719877261237764430

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