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Properties

Label 1-1375-1375.1038-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.310 - 0.950i$
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  + (−0.770 + 0.637i)2-s + (0.904 − 0.425i)3-s + (0.187 − 0.982i)4-s + (−0.425 + 0.904i)6-s + (−0.951 − 0.309i)7-s + (0.481 + 0.876i)8-s + (0.637 − 0.770i)9-s + (−0.248 − 0.968i)12-s + (−0.770 − 0.637i)13-s + (0.929 − 0.368i)14-s + (−0.929 − 0.368i)16-s + (−0.481 − 0.876i)17-s + i·18-s + (0.992 + 0.125i)19-s + (−0.992 + 0.125i)21-s + ⋯
L(s)  = 1  + (−0.770 + 0.637i)2-s + (0.904 − 0.425i)3-s + (0.187 − 0.982i)4-s + (−0.425 + 0.904i)6-s + (−0.951 − 0.309i)7-s + (0.481 + 0.876i)8-s + (0.637 − 0.770i)9-s + (−0.248 − 0.968i)12-s + (−0.770 − 0.637i)13-s + (0.929 − 0.368i)14-s + (−0.929 − 0.368i)16-s + (−0.481 − 0.876i)17-s + i·18-s + (0.992 + 0.125i)19-s + (−0.992 + 0.125i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.310 - 0.950i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1038, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.310 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8548002626 - 1.178214167i\)
\(L(\frac12)\) \(\approx\) \(0.8548002626 - 1.178214167i\)
\(L(1)\) \(\approx\) \(0.9045645309 - 0.1351619230i\)
\(L(1)\) \(\approx\) \(0.9045645309 - 0.1351619230i\)

Euler product

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

−20.73956985158440830483470556471, −19.936340053248464710340349032140, −19.381083719059421649009628567758, −18.977204424551808114956252242677, −18.05615591676204487932175755381, −17.017790969036509775604289443251, −16.44636728476778723361391705223, −15.57541204486423659420285948250, −15.02473551120823481672568561659, −13.78030018571964414560552751145, −13.19793804956725238577398693121, −12.40512394871718222341105229348, −11.534126076360404057390589130058, −10.60565134827972839594542174110, −9.73807892869667177346904003892, −9.35861615363566693205654208256, −8.68370060338189811877849904861, −7.636204759895424690614307224855, −7.093640820919680191053369987156, −5.881485445052287557449335286908, −4.44280185078918372516624034292, −3.79007518623907687504783725253, −2.76877989210001293715640212571, −2.31745166969515166314289163269, −1.05866889514301491544770878136, 0.376348754783791395949348109225, 1.13090766363149625118149043538, 2.51499287307859277659123064625, 3.068846826345035443793114554024, 4.427262385886864901695390466683, 5.49209624259723268058391936658, 6.51884264461791172038192385447, 7.306004884938666134683844261256, 7.58549415105199687948402905273, 8.89641539743996400389762308773, 9.22130085705202511914540932214, 10.06946190926244987632257791538, 10.798082506773710648230621457071, 12.10799899105332049442721123652, 12.89001878469794812354685505252, 13.735538677586587666026963044599, 14.41553633108693184837132569999, 15.11427463877820516944552163991, 16.04910053398452006667531521708, 16.392336855300943471351996947826, 17.68426231263863114109682299433, 18.05393445630724001154106941945, 18.99710051126216845982498349666, 19.57815563856768115867709689188, 20.14122746058964183533744509077

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