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Properties

Label 1-1375-1375.492-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: $\chi_{1375} (492, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ 0.866 + 0.498i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.316397546 + 0.6185969453i\)
\(L(\frac12)\) \(\approx\) \(2.316397546 + 0.6185969453i\)
\(L(1)\) \(\approx\) \(1.639048812 - 0.09688357617i\)
\(L(1)\) \(\approx\) \(1.639048812 - 0.09688357617i\)

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

−20.77104064041883061503507474945, −20.194499454284954141258282434567, −19.22673501918980629707047410237, −18.22896858411615033944501542973, −17.830578583242720126156073714525, −17.00653115407185063959872275523, −16.05498011377453780028444136902, −15.38348984495555948772291357806, −14.48269212083322328537723575675, −13.96514766924506155964283940876, −13.36470673318063842641351320541, −12.71352340803281733517324111678, −11.71379141891658704436565185066, −10.85263270173279746311317889221, −9.5631291759975639125526111439, −8.715739879554194135547026915584, −8.11279367625933306329715592205, −7.41492949292052686613639371400, −6.57554169032810757949971859249, −6.017594838416288289694852155, −4.486182956926548093630194077052, −4.17537786655689942401847843838, −3.0353909257672455986808822851, −2.04966830391577940526799980077, −0.702798639136509287296557991171, 1.482348631269649411125630236, 2.21425765998366771944660511471, 3.04964715402150312600711053785, 4.04272225006255041283095695121, 4.55830733374846770082199026153, 5.622735907646194043922532388885, 6.330070587040952626380517924548, 8.007446061877576451384742497594, 8.616324061805244409804646994180, 9.18961135601246084860498548881, 10.26824856760507981394620695217, 10.775126183740860365571025058212, 11.58807191190473827012007403037, 12.5037461546794375264835286202, 13.27797773260417728083717179060, 14.04935149222842936798399484203, 14.68362751870344663402889814661, 15.45686642885623052089840547152, 15.89985219694709401297579690157, 17.29458319556223508829047355024, 18.181725712665983370959477334119, 18.800655716399930576768158042906, 19.66857229595654492512381873462, 20.191330174680347925687260144962, 21.009666139386611766793019290977

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