Properties

Label 1-1375-1375.304-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.498 - 0.866i$
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  + (0.876 − 0.481i)2-s + (0.637 + 0.770i)3-s + (0.535 − 0.844i)4-s + (0.929 + 0.368i)6-s + (−0.809 − 0.587i)7-s + (0.0627 − 0.998i)8-s + (−0.187 + 0.982i)9-s + (0.992 − 0.125i)12-s + (−0.187 + 0.982i)13-s + (−0.992 − 0.125i)14-s + (−0.425 − 0.904i)16-s + (−0.637 + 0.770i)17-s + (0.309 + 0.951i)18-s + (0.929 + 0.368i)19-s + (−0.0627 − 0.998i)21-s + ⋯
L(s)  = 1  + (0.876 − 0.481i)2-s + (0.637 + 0.770i)3-s + (0.535 − 0.844i)4-s + (0.929 + 0.368i)6-s + (−0.809 − 0.587i)7-s + (0.0627 − 0.998i)8-s + (−0.187 + 0.982i)9-s + (0.992 − 0.125i)12-s + (−0.187 + 0.982i)13-s + (−0.992 − 0.125i)14-s + (−0.425 − 0.904i)16-s + (−0.637 + 0.770i)17-s + (0.309 + 0.951i)18-s + (0.929 + 0.368i)19-s + (−0.0627 − 0.998i)21-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.498 - 0.866i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.498 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.349420942 - 2.332749222i\)
\(L(\frac12)\) \(\approx\) \(1.349420942 - 2.332749222i\)
\(L(1)\) \(\approx\) \(1.709411747 - 0.4036549796i\)
\(L(1)\) \(\approx\) \(1.709411747 - 0.4036549796i\)

Euler product

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

−20.88366952228181781217336365773, −20.02974721886749717680476091913, −19.73647607160965789529820148500, −18.51209010990278754572447071736, −17.9665453944536742674427326474, −17.08318048001751720103243195797, −16.06148046946962342018826328846, −15.364281717984850462541158722073, −14.924190004272591496381696726044, −13.78030827030325034058206759050, −13.37493108322407342790698658289, −12.68899437971590717878676332196, −11.97880253463024435686597227863, −11.25493558708054479123471687324, −9.83155697970195067507186798554, −8.99684048458848258389451185710, −8.23168545658956450311572143634, −7.21419521184570746321301919121, −6.88936400078407969277747019958, −5.76149033516019273277625531876, −5.21123625537486186919405699936, −3.8381785114100064778038000281, −2.93798011253429323177006356771, −2.61632145778954559404688575188, −1.19305342213833148884811862491, 0.315520559850187871130238740956, 1.821482273751601362250395011023, 2.593545680627585769972607551213, 3.702891958296817297632947645115, 4.005053401017090098140731870861, 4.95159326753816712040664281795, 5.95466812620646774337322746881, 6.84041720265799833716178148267, 7.717348142540692049076868697453, 9.076398101192217973583565504113, 9.56738866370962430496880880021, 10.43263581064668504070750875638, 11.00674410763010008314751062743, 11.95583574114860092201540652576, 12.968677008816240294600411723985, 13.51571347197150324457277565240, 14.26090405808032427172265538063, 14.90534055537662693731284221414, 15.70214454785678589655943562353, 16.4227829405143791877155601292, 16.98305575801577295427408834863, 18.57669822926750369627204239875, 19.18668143043287184620747191391, 19.88913982008140579412288177798, 20.44822188136945956913781223487

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