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Properties

Label 1-1375-1375.8-r0-0-0
Degree $1$
Conductor $1375$
Sign $0.866 + 0.498i$
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  + (−0.481 + 0.876i)2-s + (−0.770 − 0.637i)3-s + (−0.535 − 0.844i)4-s + (0.929 − 0.368i)6-s + (−0.587 − 0.809i)7-s + (0.998 − 0.0627i)8-s + (0.187 + 0.982i)9-s + (−0.125 + 0.992i)12-s + (−0.982 + 0.187i)13-s + (0.992 − 0.125i)14-s + (−0.425 + 0.904i)16-s + (0.770 − 0.637i)17-s + (−0.951 − 0.309i)18-s + (−0.929 + 0.368i)19-s + (−0.0627 + 0.998i)21-s + ⋯
L(s)  = 1  + (−0.481 + 0.876i)2-s + (−0.770 − 0.637i)3-s + (−0.535 − 0.844i)4-s + (0.929 − 0.368i)6-s + (−0.587 − 0.809i)7-s + (0.998 − 0.0627i)8-s + (0.187 + 0.982i)9-s + (−0.125 + 0.992i)12-s + (−0.982 + 0.187i)13-s + (0.992 − 0.125i)14-s + (−0.425 + 0.904i)16-s + (0.770 − 0.637i)17-s + (−0.951 − 0.309i)18-s + (−0.929 + 0.368i)19-s + (−0.0627 + 0.998i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.866 + 0.498i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.866 + 0.498i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5538884787 + 0.1479166309i\)
\(L(\frac12)\) \(\approx\) \(0.5538884787 + 0.1479166309i\)
\(L(1)\) \(\approx\) \(0.5489055359 + 0.06617890351i\)
\(L(1)\) \(\approx\) \(0.5489055359 + 0.06617890351i\)

Euler product

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

−21.01478902844815847727243577308, −19.94221032801780934612722987963, −19.224435060419066434873711034, −18.647496496240564105103975831982, −17.696579153227817790332196684533, −17.03707357505956815364730983266, −16.58629139990748039170400825737, −15.438054321669201988908171194623, −14.98111184308491692328940706051, −13.68635081819940424694880958635, −12.540656474011244397141558425277, −12.34003520426902158995379929169, −11.52072850054466298595770987238, −10.53523703335792204295146136893, −10.125969856368450049731413793422, −9.21875970179216365654493802543, −8.69229982127297458137432285659, −7.47658161161508668520997398159, −6.501348276289401178222525246890, −5.46111955334114901933200855022, −4.74737386069450766285796733377, −3.71158313500543494690438068963, −2.93966478481831745221927379721, −1.91520446187723541673063467093, −0.49956164027609503046808393912, 0.65553125308516710254420151985, 1.61438989292629322263553910016, 3.015185078061117607651701453014, 4.56639940235503877092070763236, 4.995418841191027641533042129091, 6.20016079968163361379534548709, 6.69797828912209480778968472132, 7.40147752032527469791063330168, 8.08002372402226626168693382717, 9.18040648495699704338187771476, 10.19736015034260187393613831265, 10.5008796689657664066169584637, 11.67358068059918633826676096001, 12.59491784396262970985952259588, 13.274055852756828501462040546651, 14.13741034316084797396164907949, 14.75938332381573267975297599618, 15.98556636084311091805766700333, 16.491962827543333860231020560687, 17.09852890996157936634917732925, 17.669330427294227120664823026075, 18.5332157887585899927247180281, 19.416380843321973168222846452923, 19.518659460136180032251058639263, 20.89551080402724561648918589198

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