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Properties

Label 1-667-667.389-r0-0-0
Degree $1$
Conductor $667$
Sign $0.555 + 0.831i$
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  + (−0.909 + 0.415i)2-s + (−0.281 − 0.959i)3-s + (0.654 − 0.755i)4-s + (0.841 + 0.540i)5-s + (0.654 + 0.755i)6-s + (0.142 + 0.989i)7-s + (−0.281 + 0.959i)8-s + (−0.841 + 0.540i)9-s + (−0.989 − 0.142i)10-s + (−0.909 − 0.415i)11-s + (−0.909 − 0.415i)12-s + (0.142 − 0.989i)13-s + (−0.540 − 0.841i)14-s + (0.281 − 0.959i)15-s + (−0.142 − 0.989i)16-s + (−0.755 + 0.654i)17-s + ⋯
L(s)  = 1  + (−0.909 + 0.415i)2-s + (−0.281 − 0.959i)3-s + (0.654 − 0.755i)4-s + (0.841 + 0.540i)5-s + (0.654 + 0.755i)6-s + (0.142 + 0.989i)7-s + (−0.281 + 0.959i)8-s + (−0.841 + 0.540i)9-s + (−0.989 − 0.142i)10-s + (−0.909 − 0.415i)11-s + (−0.909 − 0.415i)12-s + (0.142 − 0.989i)13-s + (−0.540 − 0.841i)14-s + (0.281 − 0.959i)15-s + (−0.142 − 0.989i)16-s + (−0.755 + 0.654i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7093430869 + 0.3790422918i\)
\(L(\frac12)\) \(\approx\) \(0.7093430869 + 0.3790422918i\)
\(L(1)\) \(\approx\) \(0.7037547040 + 0.09386643491i\)
\(L(1)\) \(\approx\) \(0.7037547040 + 0.09386643491i\)

Euler product

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

−22.37548638401079595251619832939, −21.49151139609344676311926736945, −20.94611368530568998457718925555, −20.316474890929614856893193380516, −19.69509804887141366595002097890, −18.22955090259274278575768918547, −17.67781516351427746598525915312, −17.01335568995052949874110988368, −16.15008830081840284611013583511, −15.769422332179073382744461695411, −14.28380460952445631528084986572, −13.46521364416821034933122109148, −12.46638335908102544622147419557, −11.29996610974984047617672950467, −10.78829775516022021664523079629, −9.797195360924898193304699277652, −9.40788836966108237634490250899, −8.47855931431003726918692978202, −7.29028029499831173973090404127, −6.38057796337843428043579543379, −5.01521335346336954304104644105, −4.33416842666250400527902253717, −3.05318365882783533476767934255, −1.95368766742586614195324201550, −0.60584378216390524688290050055, 1.150779497209637955371864919139, 2.29504642794082740640011224360, 2.81731953002077158956977757810, 5.37083391681257505256419372653, 5.78477589539288749306645042605, 6.488031373093648674342027348223, 7.659207873650053623896234401201, 8.23270671900348564229519123191, 9.19688211829670627419160623290, 10.30096490590915125786553773031, 10.95852003538528913974121789214, 11.88098125435458806505635216557, 12.98130700059447228700075562417, 13.73307875285349527086252485525, 14.81343940243127359976890511342, 15.47277847236310816582587546218, 16.55668389135345506810448321921, 17.49070364571981864523578559288, 18.12613602678439359892783967379, 18.47709621434423181347457575967, 19.2357262677527618465776921075, 20.279715108798657839514246242518, 21.197707434765216628057496293761, 22.26218021632381874585352899953, 22.98423467535627284595567763350

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