Properties

Label 1-1375-1375.1053-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.970 + 0.243i$
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 + (0.844 + 0.535i)3-s + (0.929 − 0.368i)4-s + (0.929 + 0.368i)6-s + (0.951 + 0.309i)7-s + (0.844 − 0.535i)8-s + (0.425 + 0.904i)9-s + (0.982 + 0.187i)12-s + (−0.904 + 0.425i)13-s + (0.992 + 0.125i)14-s + (0.728 − 0.684i)16-s + (−0.770 − 0.637i)17-s + (0.587 + 0.809i)18-s + (0.968 − 0.248i)19-s + (0.637 + 0.770i)21-s + ⋯
L(s)  = 1  + (0.982 − 0.187i)2-s + (0.844 + 0.535i)3-s + (0.929 − 0.368i)4-s + (0.929 + 0.368i)6-s + (0.951 + 0.309i)7-s + (0.844 − 0.535i)8-s + (0.425 + 0.904i)9-s + (0.982 + 0.187i)12-s + (−0.904 + 0.425i)13-s + (0.992 + 0.125i)14-s + (0.728 − 0.684i)16-s + (−0.770 − 0.637i)17-s + (0.587 + 0.809i)18-s + (0.968 − 0.248i)19-s + (0.637 + 0.770i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.970 + 0.243i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.970 + 0.243i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.557462472 + 0.5621690805i\)
\(L(\frac12)\) \(\approx\) \(4.557462472 + 0.5621690805i\)
\(L(1)\) \(\approx\) \(2.728515677 + 0.1839766176i\)
\(L(1)\) \(\approx\) \(2.728515677 + 0.1839766176i\)

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 \( 1 + (0.844 + 0.535i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (-0.904 + 0.425i)T \)
17 \( 1 + (-0.770 - 0.637i)T \)
19 \( 1 + (0.968 - 0.248i)T \)
23 \( 1 + (0.125 - 0.992i)T \)
29 \( 1 + (-0.637 - 0.770i)T \)
31 \( 1 + (0.968 - 0.248i)T \)
37 \( 1 + (0.982 + 0.187i)T \)
41 \( 1 + (-0.728 + 0.684i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (-0.770 + 0.637i)T \)
53 \( 1 + (-0.368 - 0.929i)T \)
59 \( 1 + (-0.728 + 0.684i)T \)
61 \( 1 + (0.187 + 0.982i)T \)
67 \( 1 + (-0.248 - 0.968i)T \)
71 \( 1 + (-0.929 + 0.368i)T \)
73 \( 1 + (0.904 + 0.425i)T \)
79 \( 1 + (0.0627 + 0.998i)T \)
83 \( 1 + (-0.998 - 0.0627i)T \)
89 \( 1 + (0.187 + 0.982i)T \)
97 \( 1 + (-0.844 - 0.535i)T \)
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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

−20.73689530928777438800967211369, −20.142325909409508824468021763805, −19.676259711378183872967512817793, −18.63405919516983056418677102119, −17.59625833538664614777635545891, −17.18930709776230167343029167617, −15.92793236629776443424361949859, −15.15429782168615049190331297927, −14.66931364163420979487205778125, −13.893254390332076057994054976319, −13.37379057954926990729820613681, −12.490262439364667707650596457308, −11.839622631093939336999035670141, −10.98111257223736469674203602711, −10.01035515872480985734757430261, −8.89239600490062935756225161517, −7.910053236434533498823785911059, −7.49651220224465869243820841944, −6.70498272649826031593465052746, −5.59800565673743683716903815793, −4.78425176481565614217366370925, −3.86931792759675493718140463269, −3.050659335246278521232175471986, −2.08871735595717291329695451988, −1.34495119480606118859153771977, 1.42018556115816747955157882916, 2.493412250296637600866834397656, 2.84242085159191655261057198955, 4.288993663394570760446564644727, 4.60334585353555347881644630184, 5.40658257695156631741734987212, 6.62355777627931021361486913527, 7.54248324796891801621055254751, 8.21698543014317027594435028677, 9.382597093173707127572043275518, 9.95052450847093300288863821900, 11.11323995751056137345022894699, 11.53276781917303184078760695526, 12.503905586221353193189645223301, 13.491859577314764358079961335855, 14.00230542940637666287415171014, 14.823793337675725140733334624676, 15.159568738954114869694687319098, 16.08824209191193801180035088873, 16.77477168098415905969220770453, 17.94426569164247280757274509117, 18.86744167678313791612261974729, 19.66739715660623575011799429773, 20.34611488624387957201521534924, 20.90670419226029785033401283166

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