Properties

Label 1-1375-1375.1066-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.999 + 0.0251i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (−0.535 + 0.844i)2-s + (0.728 − 0.684i)3-s + (−0.425 − 0.904i)4-s + (0.187 + 0.982i)6-s + (0.809 − 0.587i)7-s + (0.992 + 0.125i)8-s + (0.0627 − 0.998i)9-s + (−0.929 − 0.368i)12-s + (−0.0627 + 0.998i)13-s + (0.0627 + 0.998i)14-s + (−0.637 + 0.770i)16-s + (0.425 − 0.904i)17-s + (0.809 + 0.587i)18-s + (−0.728 − 0.684i)19-s + (0.187 − 0.982i)21-s + ⋯
L(s)  = 1  + (−0.535 + 0.844i)2-s + (0.728 − 0.684i)3-s + (−0.425 − 0.904i)4-s + (0.187 + 0.982i)6-s + (0.809 − 0.587i)7-s + (0.992 + 0.125i)8-s + (0.0627 − 0.998i)9-s + (−0.929 − 0.368i)12-s + (−0.0627 + 0.998i)13-s + (0.0627 + 0.998i)14-s + (−0.637 + 0.770i)16-s + (0.425 − 0.904i)17-s + (0.809 + 0.587i)18-s + (−0.728 − 0.684i)19-s + (0.187 − 0.982i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.999 + 0.0251i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1066, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.999 + 0.0251i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003010496497 - 0.2395550889i\)
\(L(\frac12)\) \(\approx\) \(0.003010496497 - 0.2395550889i\)
\(L(1)\) \(\approx\) \(0.9185458994 + 0.01605978112i\)
\(L(1)\) \(\approx\) \(0.9185458994 + 0.01605978112i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.535 + 0.844i)T \)
3 \( 1 + (0.728 - 0.684i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (-0.0627 + 0.998i)T \)
17 \( 1 + (0.425 - 0.904i)T \)
19 \( 1 + (-0.728 - 0.684i)T \)
23 \( 1 + (0.968 + 0.248i)T \)
29 \( 1 + (-0.876 - 0.481i)T \)
31 \( 1 + (-0.425 + 0.904i)T \)
37 \( 1 + (-0.637 + 0.770i)T \)
41 \( 1 + (-0.968 + 0.248i)T \)
43 \( 1 + (-0.309 + 0.951i)T \)
47 \( 1 + (-0.992 + 0.125i)T \)
53 \( 1 + (-0.187 + 0.982i)T \)
59 \( 1 + (-0.929 - 0.368i)T \)
61 \( 1 + (-0.968 - 0.248i)T \)
67 \( 1 + (0.876 - 0.481i)T \)
71 \( 1 + (-0.992 + 0.125i)T \)
73 \( 1 + (0.929 - 0.368i)T \)
79 \( 1 + (-0.728 + 0.684i)T \)
83 \( 1 + (-0.728 - 0.684i)T \)
89 \( 1 + (-0.929 + 0.368i)T \)
97 \( 1 + (0.876 + 0.481i)T \)
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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

−20.964088795241492159542544554381, −20.389539684203814167415063075731, −19.60247941090349774784508036550, −18.82859746946462871899510253660, −18.30193657529574273547599541016, −17.15557137867779267519398880171, −16.80369491105466163482224018886, −15.59328912609208898690171986773, −14.89979740538255753898683682589, −14.310759891226924889957465933044, −13.12718618187902465161274108528, −12.64844773956305301988011744645, −11.60284963580772928221689471956, −10.71859178567863295511062357448, −10.33746772850025138243838573688, −9.34156528578612115349399355997, −8.57408877008094056492450394103, −8.13523025680255291483070766311, −7.31304105771971449315903620133, −5.64738101938406719357832291560, −4.88580877280086239303574681121, −3.862850665813661627199796507508, −3.216964176470503115784457277985, −2.17882294063017014861000029096, −1.54031351470501659643410079480, 0.04700904304262043029316088945, 1.24712436097136330516783424634, 1.837632014294365340528803622971, 3.16801668150682963967249265426, 4.41918405667156984766953067743, 5.083189266255000705964662934935, 6.39717491848738661621301087182, 7.032718912883328755099905054, 7.60225029076972308628285423102, 8.44471422921467147362821514886, 9.11069328011492454139757285757, 9.82261558113991249747399297565, 10.97423283621735569621097744497, 11.65648403076064101430440784031, 12.902020188712625284074633681681, 13.71600336235831266783744162936, 14.17445348734109799450585344163, 14.90502054604269111450206091455, 15.56463985852751154710739291895, 16.76734935472218168574170952991, 17.14744382450682896569943986412, 18.11568431397361892573634258673, 18.6244470905165864287694016909, 19.368649722953340209633374086375, 20.05424723934485969132088925195

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