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Properties

Label 1-4675-4675.2066-r0-0-0
Degree $1$
Conductor $4675$
Sign $0.614 + 0.788i$
Analytic cond. $21.7106$
Root an. cond. $21.7106$
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  + (−0.951 − 0.309i)2-s + (−0.453 + 0.891i)3-s + (0.809 + 0.587i)4-s + (0.707 − 0.707i)6-s + (−0.891 − 0.453i)7-s + (−0.587 − 0.809i)8-s + (−0.587 − 0.809i)9-s + (−0.891 + 0.453i)12-s + (0.809 − 0.587i)13-s + (0.707 + 0.707i)14-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)18-s + (0.587 + 0.809i)19-s + (0.809 − 0.587i)21-s + (0.891 + 0.453i)23-s + (0.987 − 0.156i)24-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.453 + 0.891i)3-s + (0.809 + 0.587i)4-s + (0.707 − 0.707i)6-s + (−0.891 − 0.453i)7-s + (−0.587 − 0.809i)8-s + (−0.587 − 0.809i)9-s + (−0.891 + 0.453i)12-s + (0.809 − 0.587i)13-s + (0.707 + 0.707i)14-s + (0.309 + 0.951i)16-s + (0.309 + 0.951i)18-s + (0.587 + 0.809i)19-s + (0.809 − 0.587i)21-s + (0.891 + 0.453i)23-s + (0.987 − 0.156i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4675\)    =    \(5^{2} \cdot 11 \cdot 17\)
Sign: $0.614 + 0.788i$
Analytic conductor: \(21.7106\)
Root analytic conductor: \(21.7106\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4675} (2066, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4675,\ (0:\ ),\ 0.614 + 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7314109978 + 0.3574271337i\)
\(L(\frac12)\) \(\approx\) \(0.7314109978 + 0.3574271337i\)
\(L(1)\) \(\approx\) \(0.5955461069 + 0.09521141561i\)
\(L(1)\) \(\approx\) \(0.5955461069 + 0.09521141561i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (-0.453 + 0.891i)T \)
7 \( 1 + (-0.891 - 0.453i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (0.587 + 0.809i)T \)
23 \( 1 + (0.891 + 0.453i)T \)
29 \( 1 + (0.156 + 0.987i)T \)
31 \( 1 + (-0.453 + 0.891i)T \)
37 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (0.987 - 0.156i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 - iT \)
59 \( 1 + iT \)
61 \( 1 + (0.987 - 0.156i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.891 - 0.453i)T \)
73 \( 1 + (0.987 + 0.156i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−18.09130867026253456957073526212, −17.53076192757645011608319194582, −16.7705234044852136487089033965, −16.26567122059801182171806935779, −15.66698300533173182526800608724, −14.918119199456765109961690603777, −13.97674971424371564817208844054, −13.34460121641219892356598561434, −12.60036431847971730747599259705, −11.92212381636741798108782411129, −11.101759171596578044836075223529, −10.87551003176873421201248902555, −9.57558741565222234303436991420, −9.29152005100102023213495962353, −8.42819404988337017846745209890, −7.753927944867294960663890923213, −6.936198126234423383084984616774, −6.515210038997815556930896130320, −5.84027550810476534779076946908, −5.23476171091049565267031484707, −3.97079785776235635746352221121, −2.68862820454557450043060351278, −2.370284336312851922344469717818, −1.20975923165501748410175418242, −0.53187718710690650381596147420, 0.71449793294393308715906281201, 1.43483537863582399231168269308, 2.90330221477204888631499849848, 3.34829502838758982228829754131, 3.92408745066994286312616782652, 5.0736805944935172283786986540, 5.898804185113741746286405385614, 6.5507845246674498417009720096, 7.27083029577231255309805096344, 8.18021551940581280378355148992, 8.90876904055422913598704678587, 9.555127136653340535424604933039, 10.08626492003640675530129898745, 10.74009531789002422828978000315, 11.179528880602488215390301736506, 12.037528449729312000278202450454, 12.71463652513731532966219583277, 13.37794290454140310977390574083, 14.46699605710437814162995222964, 15.22037857138508519207909761795, 15.91850030740813052642527265004, 16.41870292581637965391150630331, 16.73316485850916221319902227458, 17.73458089193677364629792671968, 18.080414580287491643798440893034

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