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Properties

Label 1-667-667.72-r1-0-0
Degree $1$
Conductor $667$
Sign $-0.120 - 0.992i$
Analytic cond. $71.6791$
Root an. cond. $71.6791$
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  + (0.872 − 0.488i)2-s + (0.965 − 0.262i)3-s + (0.523 − 0.852i)4-s + (0.933 − 0.359i)5-s + (0.714 − 0.699i)6-s + (−0.768 + 0.639i)7-s + (0.0407 − 0.999i)8-s + (0.862 − 0.505i)9-s + (0.639 − 0.768i)10-s + (0.574 + 0.818i)11-s + (0.281 − 0.959i)12-s + (−0.970 − 0.242i)13-s + (−0.359 + 0.933i)14-s + (0.806 − 0.591i)15-s + (−0.452 − 0.891i)16-s + (0.540 − 0.841i)17-s + ⋯
L(s)  = 1  + (0.872 − 0.488i)2-s + (0.965 − 0.262i)3-s + (0.523 − 0.852i)4-s + (0.933 − 0.359i)5-s + (0.714 − 0.699i)6-s + (−0.768 + 0.639i)7-s + (0.0407 − 0.999i)8-s + (0.862 − 0.505i)9-s + (0.639 − 0.768i)10-s + (0.574 + 0.818i)11-s + (0.281 − 0.959i)12-s + (−0.970 − 0.242i)13-s + (−0.359 + 0.933i)14-s + (0.806 − 0.591i)15-s + (−0.452 − 0.891i)16-s + (0.540 − 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.120 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $-0.120 - 0.992i$
Analytic conductor: \(71.6791\)
Root analytic conductor: \(71.6791\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (1:\ ),\ -0.120 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.141053689 - 4.673902601i\)
\(L(\frac12)\) \(\approx\) \(4.141053689 - 4.673902601i\)
\(L(1)\) \(\approx\) \(2.431729755 - 1.361395251i\)
\(L(1)\) \(\approx\) \(2.431729755 - 1.361395251i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.872 - 0.488i)T \)
3 \( 1 + (0.965 - 0.262i)T \)
5 \( 1 + (0.933 - 0.359i)T \)
7 \( 1 + (-0.768 + 0.639i)T \)
11 \( 1 + (0.574 + 0.818i)T \)
13 \( 1 + (-0.970 - 0.242i)T \)
17 \( 1 + (0.540 - 0.841i)T \)
19 \( 1 + (0.852 + 0.523i)T \)
31 \( 1 + (0.470 - 0.882i)T \)
37 \( 1 + (-0.505 - 0.862i)T \)
41 \( 1 + (0.989 - 0.142i)T \)
43 \( 1 + (-0.470 - 0.882i)T \)
47 \( 1 + (0.781 + 0.623i)T \)
53 \( 1 + (-0.979 - 0.202i)T \)
59 \( 1 + (0.415 + 0.909i)T \)
61 \( 1 + (-0.396 - 0.917i)T \)
67 \( 1 + (0.818 + 0.574i)T \)
71 \( 1 + (0.377 - 0.925i)T \)
73 \( 1 + (-0.940 + 0.339i)T \)
79 \( 1 + (0.891 + 0.452i)T \)
83 \( 1 + (-0.301 - 0.953i)T \)
89 \( 1 + (0.806 + 0.591i)T \)
97 \( 1 + (-0.830 + 0.557i)T \)
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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

−22.572163200468926015055050431561, −21.88605620231487553839059833824, −21.48126875878745098714940349083, −20.49371564194780388415105589086, −19.67035255942244449144249492765, −18.98524153852024539763636413426, −17.59180581511890623489309399389, −16.805554567728277535221376349850, −16.13947502105811469433084371288, −15.11827307813085363294404567720, −14.267715498387925491741576697080, −13.88637784364052126836150991486, −13.1522764421047396880067787474, −12.28306246658034213610528101230, −10.94607431318173195208932062141, −9.9978886256944678059639538401, −9.24749208635643492675988620812, −8.16345005295413612918070387914, −7.116858667263407513142960450640, −6.512367086123080799845102730880, −5.42735734972398797929337835424, −4.338918089525964824053438705576, −3.28828554610625610873987645374, −2.81236830975199148057048666183, −1.475116972242817871421040199349, 0.93409740370763124435614528441, 2.11093563323628587972148684439, 2.65510089646762514566604067123, 3.68870124357982420625316313943, 4.86112168292470808928077629500, 5.74548274505366298756575359833, 6.73676265408325498368737350261, 7.598324595626913193971001245079, 9.2685536121199639819508848438, 9.53219494600759907239305199430, 10.25762908757367352824580897489, 12.0195829317597815223043561717, 12.377866387728317350927980472822, 13.17536303970373278245284840496, 14.04179212057372451879548818529, 14.56787178189534785694388387205, 15.47640771628716962539603272500, 16.32198552621594864009496540866, 17.58472530615144377626370313377, 18.59373671098500681046568940628, 19.26505523167417073275788461976, 20.2001979629159244095762912649, 20.613866128502749178939271867619, 21.524179883392056108970967415937, 22.29475760958623794033822728040

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