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Properties

Label 1-667-667.505-r0-0-0
Degree $1$
Conductor $667$
Sign $0.981 + 0.189i$
Analytic cond. $3.09753$
Root an. cond. $3.09753$
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  + i·2-s + i·3-s − 4-s + 5-s − 6-s − 7-s − i·8-s − 9-s + i·10-s − i·11-s − i·12-s − 13-s − i·14-s + i·15-s + 16-s − i·17-s + ⋯
L(s)  = 1  + i·2-s + i·3-s − 4-s + 5-s − 6-s − 7-s − i·8-s − 9-s + i·10-s − i·11-s − i·12-s − 13-s − i·14-s + i·15-s + 16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.981 + 0.189i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (505, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.981 + 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8108793959 + 0.07736971083i\)
\(L(\frac12)\) \(\approx\) \(0.8108793959 + 0.07736971083i\)
\(L(1)\) \(\approx\) \(0.7309553120 + 0.4312374957i\)
\(L(1)\) \(\approx\) \(0.7309553120 + 0.4312374957i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + iT \)
7 \( 1 + iT \)
11 \( 1 - T \)
13 \( 1 + T \)
17 \( 1 - T \)
19 \( 1 - T \)
31 \( 1 + iT \)
37 \( 1 - iT \)
41 \( 1 - iT \)
43 \( 1 - T \)
47 \( 1 - iT \)
53 \( 1 + iT \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 - iT \)
71 \( 1 - iT \)
73 \( 1 - T \)
79 \( 1 - iT \)
83 \( 1 + T \)
89 \( 1 \)
97 \( 1 + T \)
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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

−22.62105661394802053557519298642, −22.04874003239116122310746702666, −21.03312022549100648044129149145, −20.17070601093089850897728323969, −19.4710898453861183595908645944, −18.82496363834496758256558802593, −17.93795582321466718211888142542, −17.32725600369283996200639298932, −16.63177072563057376463520040744, −14.8109239822990018006560165193, −14.27554044398879043621168506264, −13.16692119474757985316778923226, −12.73060070237710945135147790219, −12.227561338282960125966453171489, −10.99858126005272590609870598897, −9.84105617136688447154542508359, −9.63194882367821756369889538879, −8.37872739775166369523104679124, −7.331761268573468725478337064334, −6.23384333957344343025693756960, −5.513862634067116338846960879893, −4.20631563070802854572697903747, −2.88230656248084700226972428506, −2.18643803354916533602336758384, −1.31608443891648038273855235523, 0.40751657777873209595291888245, 2.68813625710026295570567260251, 3.49046740956213858103140336709, 4.873483126102717764441253934457, 5.38079542384349476974060037630, 6.354818180537663423695912001661, 7.10929721653733219189189693937, 8.608866820558506756359996631864, 9.21727021206244204609896077189, 9.86331622812750725759461548001, 10.60899258318968254003130712937, 12.00578349397294237886461236541, 13.22520476168425258987001704355, 13.84412954003926614794265785297, 14.54321580689621799567417451503, 15.60949655242813181072782160550, 16.14341076093685357181756799422, 16.96045312534504384698977525387, 17.45594242968185980265470454719, 18.58528373190607110539618992393, 19.45000701123682840970875116381, 20.48305330797543061882254290637, 21.678424372348111996912311501611, 21.99569876771162554483133209670, 22.5871985014063194314873732479

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