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Properties

Label 1-1375-1375.1128-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.489 - 0.871i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (0.982 − 0.187i)2-s + (−0.368 − 0.929i)3-s + (0.929 − 0.368i)4-s + (−0.535 − 0.844i)6-s + i·7-s + (0.844 − 0.535i)8-s + (−0.728 + 0.684i)9-s + (−0.684 − 0.728i)12-s + (−0.684 − 0.728i)13-s + (0.187 + 0.982i)14-s + (0.728 − 0.684i)16-s + (0.248 + 0.968i)17-s + (−0.587 + 0.809i)18-s + (0.968 − 0.248i)19-s + (0.929 − 0.368i)21-s + ⋯
L(s)  = 1  + (0.982 − 0.187i)2-s + (−0.368 − 0.929i)3-s + (0.929 − 0.368i)4-s + (−0.535 − 0.844i)6-s + i·7-s + (0.844 − 0.535i)8-s + (−0.728 + 0.684i)9-s + (−0.684 − 0.728i)12-s + (−0.684 − 0.728i)13-s + (0.187 + 0.982i)14-s + (0.728 − 0.684i)16-s + (0.248 + 0.968i)17-s + (−0.587 + 0.809i)18-s + (0.968 − 0.248i)19-s + (0.929 − 0.368i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.489 - 0.871i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.489 - 0.871i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.315208566 - 1.354533798i\)
\(L(\frac12)\) \(\approx\) \(2.315208566 - 1.354533798i\)
\(L(1)\) \(\approx\) \(1.658545238 - 0.6007567269i\)
\(L(1)\) \(\approx\) \(1.658545238 - 0.6007567269i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.982 - 0.187i)T \)
3 \( 1 + (-0.368 - 0.929i)T \)
7 \( 1 + iT \)
13 \( 1 + (-0.684 - 0.728i)T \)
17 \( 1 + (0.248 + 0.968i)T \)
19 \( 1 + (0.968 - 0.248i)T \)
23 \( 1 + (0.481 + 0.876i)T \)
29 \( 1 + (-0.637 - 0.770i)T \)
31 \( 1 + (0.0627 - 0.998i)T \)
37 \( 1 + (0.481 - 0.876i)T \)
41 \( 1 + (0.187 - 0.982i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (0.998 - 0.0627i)T \)
53 \( 1 + (0.844 + 0.535i)T \)
59 \( 1 + (0.425 + 0.904i)T \)
61 \( 1 + (0.992 + 0.125i)T \)
67 \( 1 + (0.844 - 0.535i)T \)
71 \( 1 + (-0.637 - 0.770i)T \)
73 \( 1 + (-0.481 - 0.876i)T \)
79 \( 1 + (-0.637 - 0.770i)T \)
83 \( 1 + (0.770 + 0.637i)T \)
89 \( 1 + (-0.728 - 0.684i)T \)
97 \( 1 + (0.998 - 0.0627i)T \)
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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

−21.003002658313484390286653315500, −20.38914685151294912484663806849, −19.988660379751809398592745293, −18.76273286861917499138930957145, −17.57630266660900121090070856842, −16.846681700468184802325987914260, −16.31359300355263154121267848498, −15.79312906759050665560208642497, −14.52410106355453479420492798379, −14.41784873444387556396042249136, −13.48687578440715611110030923449, −12.47599333482382188353690361480, −11.684434597225008521231920754827, −11.09809340491721977709354858215, −10.22926389585265061503759192497, −9.557604600725783728116507505957, −8.39482334704568868329967503018, −7.165337046950759528200569429638, −6.829733441543016192434366215051, −5.571877764293094117965207890290, −4.92061462861493289839154491771, −4.255033379615646224658313026525, −3.43066219578153360343973283005, −2.60654598473916615471233935294, −1.06550046021428284233622088722, 0.93581185833670428532653571323, 2.10465146684219109454008932282, 2.655382146541357330842585075105, 3.71437499369923561993946165627, 4.99105996889129226612825520853, 5.74471724500890543532774574367, 6.03700223204758329186906767292, 7.406085055003262031594240868654, 7.67909556810048843169422296947, 8.991708803678834835843296224242, 10.04214184873933370144060600665, 11.05667983826983996727675534504, 11.708707070634369559354662743437, 12.34475379490877153430156922436, 12.96894569936913368103648820264, 13.597735456678976182065978375038, 14.61715009384098008845916766504, 15.17411196023783847632669314432, 15.985660669649829451924951672495, 16.9974231169028645932590424303, 17.67459923745887162622051998130, 18.63005977245723418955926134038, 19.30792117958718608331252454901, 19.84894049912014724050096568396, 20.84080381467418200884463900065

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