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Properties

Label 1-667-667.452-r0-0-0
Degree $1$
Conductor $667$
Sign $-0.699 - 0.714i$
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  + (−0.281 + 0.959i)2-s + (0.755 + 0.654i)3-s + (−0.841 − 0.540i)4-s + (−0.142 + 0.989i)5-s + (−0.841 + 0.540i)6-s + (−0.415 − 0.909i)7-s + (0.755 − 0.654i)8-s + (0.142 + 0.989i)9-s + (−0.909 − 0.415i)10-s + (−0.281 − 0.959i)11-s + (−0.281 − 0.959i)12-s + (−0.415 + 0.909i)13-s + (0.989 − 0.142i)14-s + (−0.755 + 0.654i)15-s + (0.415 + 0.909i)16-s + (0.540 + 0.841i)17-s + ⋯
L(s)  = 1  + (−0.281 + 0.959i)2-s + (0.755 + 0.654i)3-s + (−0.841 − 0.540i)4-s + (−0.142 + 0.989i)5-s + (−0.841 + 0.540i)6-s + (−0.415 − 0.909i)7-s + (0.755 − 0.654i)8-s + (0.142 + 0.989i)9-s + (−0.909 − 0.415i)10-s + (−0.281 − 0.959i)11-s + (−0.281 − 0.959i)12-s + (−0.415 + 0.909i)13-s + (0.989 − 0.142i)14-s + (−0.755 + 0.654i)15-s + (0.415 + 0.909i)16-s + (0.540 + 0.841i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $-0.699 - 0.714i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (452, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ -0.699 - 0.714i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2776544531 + 0.6607277119i\)
\(L(\frac12)\) \(\approx\) \(-0.2776544531 + 0.6607277119i\)
\(L(1)\) \(\approx\) \(0.5345201703 + 0.6376511734i\)
\(L(1)\) \(\approx\) \(0.5345201703 + 0.6376511734i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.281 + 0.959i)T \)
3 \( 1 + (0.755 + 0.654i)T \)
5 \( 1 + (-0.142 + 0.989i)T \)
7 \( 1 + (-0.415 - 0.909i)T \)
11 \( 1 + (-0.281 - 0.959i)T \)
13 \( 1 + (-0.415 + 0.909i)T \)
17 \( 1 + (0.540 + 0.841i)T \)
19 \( 1 + (-0.540 + 0.841i)T \)
31 \( 1 + (-0.755 + 0.654i)T \)
37 \( 1 + (-0.989 + 0.142i)T \)
41 \( 1 + (-0.989 - 0.142i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (0.415 - 0.909i)T \)
61 \( 1 + (0.755 - 0.654i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (0.959 + 0.281i)T \)
73 \( 1 + (-0.540 + 0.841i)T \)
79 \( 1 + (0.909 + 0.415i)T \)
83 \( 1 + (0.142 + 0.989i)T \)
89 \( 1 + (-0.755 - 0.654i)T \)
97 \( 1 + (0.989 + 0.142i)T \)
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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

−22.124828055627959063476879207490, −21.10905495766828454851544751831, −20.472341675121492829172957385847, −19.8691258081699532601799583291, −19.21101906205848700181159140149, −18.27763381921960968399729172467, −17.73186867018966159787064119896, −16.7205649474567604476094103513, −15.5009622052730795257194354328, −14.796920038155118007391249948250, −13.42908023744316898720797846098, −12.98096827848764506492090632528, −12.2391496627590673990275209400, −11.77326857264567697754504419765, −10.18263518913454945703890074936, −9.415411764147867872845196127338, −8.804607128384190695089028555, −7.981988334298542738532825902629, −7.13856538165857254483398874812, −5.50661898235717467093362664333, −4.66402086600267633137707978310, −3.386800712544718149056411937615, −2.52036683229749923727710893410, −1.702041766759442549908174833416, −0.33691558545039987737658539799, 1.80950010084447335138682597008, 3.46580306800937271560069500643, 3.816590659215129076169625564281, 5.09193146174224303721727434346, 6.28621082696123502736121718006, 7.06459048278279763402586982211, 7.947399185095390427273983640147, 8.66380281312877326059259758575, 9.85184356351267667408457636072, 10.31869177142654054420503218162, 11.10089026633052557259374752342, 12.81122559741998118260503123450, 13.95771161699509688971796919534, 14.171424580900686389898977807605, 14.99124206080049532129504603876, 15.90467750255292726864114764490, 16.58501250448651701113149364738, 17.230497116896594070821262397455, 18.64044302693766823826832394574, 19.10654927921456529696787426511, 19.67475529898982703957627758133, 20.97382513951974127248564125343, 21.84154573695233389416298955365, 22.46115396366120953204273165362, 23.52305446337264203032919753537

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