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Properties

Label 1-1375-1375.587-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: $\chi_{1375} (587, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.310 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.643023379 + 1.192021666i\)
\(L(\frac12)\) \(\approx\) \(1.643023379 + 1.192021666i\)
\(L(1)\) \(\approx\) \(1.305499606 - 0.1311079764i\)
\(L(1)\) \(\approx\) \(1.305499606 - 0.1311079764i\)

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

−20.770179025563778180570531854674, −19.98483247853550134471457831471, −18.60212770624210277203737954674, −17.94412291013888213314463634186, −17.55586838543873804320044336209, −16.52981683410397237160245028845, −16.08448431479263474711219355, −15.23318097205029409953560136373, −14.21478290412533480891074810613, −13.68579502922596591714489863465, −12.91528729773653812200950897665, −12.010725429139318393722099575082, −11.47708107079363023006197847113, −10.78989866303312823992868510977, −9.671306762470707448728239762913, −8.23529577605798211811029832619, −7.7965014071780233044039897701, −7.005028874423382946214139851238, −6.07454405964593950529140160070, −5.40802646147548036682840009634, −4.72333583700179051196871352236, −3.84607364506384962492025201611, −2.641988851988750793420782442485, −1.459389624336169907760772275542, −0.332318673223854757372378786938, 1.29323476618237713392461427963, 1.643528230403279490051315144366, 3.2570285246715440361457245736, 3.9485915989843529700622357764, 4.88730624944814164511595357746, 5.466833082085726651888140711523, 6.21002526705604113483888095114, 7.13430314044034458319778406124, 8.42199731006652436888423400741, 9.40004972174820640724304146641, 10.26697821319517246128344242113, 10.92925299820827109815411106570, 11.63187206493742434795583229280, 12.1121674680288943646432160101, 12.94709519961449381253897008886, 14.06736383658100171801729501075, 14.4960123758509477407332734122, 15.57643341195872343437153476483, 15.984931350894899682808971516974, 17.007888603644815516782212281354, 18.035863848486399966760379344102, 18.36554227613727965614145190033, 19.39171561041232970397661446747, 20.36110591247622365937507217725, 21.02503992707864345793646954174

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