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Properties

Label 1-667-667.360-r0-0-0
Degree $1$
Conductor $667$
Sign $-0.384 - 0.923i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.384 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $-0.384 - 0.923i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (360, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ -0.384 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4842987588 - 0.7263379465i\)
\(L(\frac12)\) \(\approx\) \(0.4842987588 - 0.7263379465i\)
\(L(1)\) \(\approx\) \(0.6526406931 - 0.4579949109i\)
\(L(1)\) \(\approx\) \(0.6526406931 - 0.4579949109i\)

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

−22.97379716041349789327135635041, −22.32837138409337128023982998212, −21.62602400909655354579347839519, −21.00110954109701114773009278823, −19.78855451804649869151774994081, −18.73051999320012921397162439759, −17.72979370856814820662623358872, −17.08569711584014895800818581130, −16.30188458876772269066526541123, −15.75584987933459183018060264950, −15.12793437108203610556371094126, −14.054392494060762980216816511787, −12.79323986630531172633515911305, −12.44205448963050278927998502277, −11.50863148364497447515122213892, −10.18808224487852804086916902293, −9.22913448231475636289062167009, −8.64012918140066282514079388551, −7.69453045692280961372752958139, −6.08531395688363063049339949040, −5.89849683287375430032028193784, −4.96232381306327151063167742291, −4.06078641153072278176057527404, −3.07751258454279146980274820130, −0.860303687301985090233630756730, 0.62180300344075739549181024101, 1.995272967730331454658844257978, 2.80804238067015101506285561017, 4.17102143910934337148687371902, 4.80041581289205783447662245628, 6.23243120852678926967375876755, 6.94197312090407875325733412537, 7.657188332172847095019577879964, 9.42521123638812041386184910873, 9.99400598908923610413104311341, 11.07215038645415204650980335435, 11.46522488114502944869850486639, 12.343012084186250579403817634168, 13.33680161466473159619314442067, 13.89715400557669945493239559324, 14.76734910302864845918074657078, 15.95954947282677639436191252749, 17.12239640881869532724234582979, 17.85000160306696911074481381131, 18.43110234260258679997360312404, 19.52655940275069239531686719431, 19.70577075402364874115450029278, 20.9177883306985669447392338935, 22.0649521544615888278982007376, 22.5028166299969856776926977677

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