Properties

Label 1-1375-1375.1164-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.0708 + 0.997i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (0.929 + 0.368i)2-s + (0.992 + 0.125i)3-s + (0.728 + 0.684i)4-s + (0.876 + 0.481i)6-s + (−0.309 + 0.951i)7-s + (0.425 + 0.904i)8-s + (0.968 + 0.248i)9-s + (0.637 + 0.770i)12-s + (−0.968 − 0.248i)13-s + (−0.637 + 0.770i)14-s + (0.0627 + 0.998i)16-s + (0.992 − 0.125i)17-s + (0.809 + 0.587i)18-s + (0.876 + 0.481i)19-s + (−0.425 + 0.904i)21-s + ⋯
L(s)  = 1  + (0.929 + 0.368i)2-s + (0.992 + 0.125i)3-s + (0.728 + 0.684i)4-s + (0.876 + 0.481i)6-s + (−0.309 + 0.951i)7-s + (0.425 + 0.904i)8-s + (0.968 + 0.248i)9-s + (0.637 + 0.770i)12-s + (−0.968 − 0.248i)13-s + (−0.637 + 0.770i)14-s + (0.0627 + 0.998i)16-s + (0.992 − 0.125i)17-s + (0.809 + 0.587i)18-s + (0.876 + 0.481i)19-s + (−0.425 + 0.904i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0708 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0708 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.0708 + 0.997i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.0708 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.740169553 + 2.941652851i\)
\(L(\frac12)\) \(\approx\) \(2.740169553 + 2.941652851i\)
\(L(1)\) \(\approx\) \(2.193607984 + 1.142800031i\)
\(L(1)\) \(\approx\) \(2.193607984 + 1.142800031i\)

Euler product

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

−20.57003247485003772530357086782, −20.00106211236982948103212383566, −19.44361709111778902424892787976, −18.77085441311590961513506408000, −17.71771342105662606920312970045, −16.45129572785937976773037634129, −16.07680851709652016435829373024, −14.927273686445715050510870880359, −14.29721522932860574940401096219, −13.98642364677695057328087459177, −12.90118260374355467642084313384, −12.55974355200554042308256672856, −11.54285720263978542523634428062, −10.43053362259066641878971730881, −9.96627045869644328919754355708, −9.118989341261315774780725011155, −7.850623584662546864257283787502, −7.14357283479095157543468313497, −6.58705760393217192105071631237, −5.19010961331173770206318625152, −4.51223323258229706572055285891, −3.47356134262724182231848704503, −3.05056935989812335964604994873, −1.93394212574438771363470964628, −0.99167625646207697764746390828, 1.71774934860625893795535792758, 2.58624986823700263996002908106, 3.26809746186420498241773867802, 4.04778425100632383481754186197, 5.23689531567227362743062534913, 5.69198151931859108758393218996, 6.933950301367253013733292775400, 7.70567083626350100988496770062, 8.23822194036870483209132005610, 9.47654447060621040410914000840, 9.87095346203748077276833863638, 11.26447720346838204649804134577, 12.16124322629450457641951021194, 12.64232951946270496218673096224, 13.54467734960941703637390802679, 14.25893997248375830162946252290, 14.88737874298841134908846765295, 15.537124187877031937243659084763, 16.12378296363433377836917775910, 17.03329307033593421692945368829, 18.04938883375824910279613934317, 19.06451016947384379006305912114, 19.54413594719677651879476621439, 20.58735513667172993304622466008, 21.00228145384673604860331388958

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