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Properties

Label 1-1375-1375.294-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: $\chi_{1375} (294, \cdot )$
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

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

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