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Properties

Label 1-667-667.247-r0-0-0
Degree $1$
Conductor $667$
Sign $-0.953 + 0.301i$
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  + (−0.806 − 0.591i)2-s + (0.852 − 0.523i)3-s + (0.301 + 0.953i)4-s + (−0.979 − 0.202i)5-s + (−0.996 − 0.0815i)6-s + (−0.742 − 0.670i)7-s + (0.320 − 0.947i)8-s + (0.452 − 0.891i)9-s + (0.670 + 0.742i)10-s + (0.983 − 0.182i)11-s + (0.755 + 0.654i)12-s + (0.377 − 0.925i)13-s + (0.202 + 0.979i)14-s + (−0.940 + 0.339i)15-s + (−0.818 + 0.574i)16-s + (−0.989 + 0.142i)17-s + ⋯
L(s)  = 1  + (−0.806 − 0.591i)2-s + (0.852 − 0.523i)3-s + (0.301 + 0.953i)4-s + (−0.979 − 0.202i)5-s + (−0.996 − 0.0815i)6-s + (−0.742 − 0.670i)7-s + (0.320 − 0.947i)8-s + (0.452 − 0.891i)9-s + (0.670 + 0.742i)10-s + (0.983 − 0.182i)11-s + (0.755 + 0.654i)12-s + (0.377 − 0.925i)13-s + (0.202 + 0.979i)14-s + (−0.940 + 0.339i)15-s + (−0.818 + 0.574i)16-s + (−0.989 + 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(667\)    =    \(23 \cdot 29\)
Sign: $-0.953 + 0.301i$
Analytic conductor: \(3.09753\)
Root analytic conductor: \(3.09753\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{667} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 667,\ (0:\ ),\ -0.953 + 0.301i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09458382793 - 0.6121964370i\)
\(L(\frac12)\) \(\approx\) \(-0.09458382793 - 0.6121964370i\)
\(L(1)\) \(\approx\) \(0.5339138311 - 0.4486935362i\)
\(L(1)\) \(\approx\) \(0.5339138311 - 0.4486935362i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.806 - 0.591i)T \)
3 \( 1 + (0.852 - 0.523i)T \)
5 \( 1 + (-0.979 - 0.202i)T \)
7 \( 1 + (-0.742 - 0.670i)T \)
11 \( 1 + (0.983 - 0.182i)T \)
13 \( 1 + (0.377 - 0.925i)T \)
17 \( 1 + (-0.989 + 0.142i)T \)
19 \( 1 + (-0.953 + 0.301i)T \)
31 \( 1 + (0.699 - 0.714i)T \)
37 \( 1 + (-0.891 - 0.452i)T \)
41 \( 1 + (-0.909 - 0.415i)T \)
43 \( 1 + (0.699 + 0.714i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (0.0611 - 0.998i)T \)
59 \( 1 + (-0.959 + 0.281i)T \)
61 \( 1 + (0.122 + 0.992i)T \)
67 \( 1 + (0.182 - 0.983i)T \)
71 \( 1 + (0.999 - 0.0407i)T \)
73 \( 1 + (-0.359 + 0.933i)T \)
79 \( 1 + (-0.574 + 0.818i)T \)
83 \( 1 + (0.768 + 0.639i)T \)
89 \( 1 + (-0.940 - 0.339i)T \)
97 \( 1 + (-0.999 + 0.0203i)T \)
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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

−23.34099313274521581862628225610, −22.44603041941805792686399065007, −21.63839693289187697408672976971, −20.41273880108625217286671014293, −19.717282136098455090157099970155, −19.12090527840379643121708723226, −18.699261027017633037074698587365, −17.35539339703217769250554171004, −16.40613193590452115568447539997, −15.72780920739285823810299858214, −15.24564154890419517213712685552, −14.453648455693775932608598101186, −13.56448255438793380353292863065, −12.205536817398070381418190672698, −11.26329899263444011137803823044, −10.403626915555977713090278745929, −9.25925939599847922796943344460, −8.88594326704457308371064642861, −8.13552803559733565342701170338, −6.858432491088612508782141771256, −6.49856469124116416845508303016, −4.82647729101417039194535285666, −3.99340888598442094408279410475, −2.824466834635516287553314700134, −1.713864060764743320161171773275, 0.362783925767466287975511928820, 1.453234753665602303267603863856, 2.764679482324460090202130569030, 3.70635320833717130163556826114, 4.14369455385088339983271564913, 6.46084668574455985279296877756, 7.03900028857249127771016148648, 8.081609925783510249291342156112, 8.58518888140453171317254368115, 9.467133044258782004423682378372, 10.46614608551322827635786323884, 11.36178399301224036222376155997, 12.36824394622162785312211180881, 12.93470360848584495205218410317, 13.751094715949434356136473758615, 15.07161940184401316668265778425, 15.74196967800677929016639912587, 16.73434378876584860554268070055, 17.506175224945796028751716076167, 18.52820120287435023246913976656, 19.45326712228185355361951264272, 19.61317055058349439851198669194, 20.34379321504626291440085635119, 21.096997100151763177489803924493, 22.411172888393651828562778137070

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