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Properties

Label 1-1375-1375.1086-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.508 - 0.860i$
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  + (−0.0627 + 0.998i)2-s + (0.728 + 0.684i)3-s + (−0.992 − 0.125i)4-s + (−0.728 + 0.684i)6-s + (−0.309 − 0.951i)7-s + (0.187 − 0.982i)8-s + (0.0627 + 0.998i)9-s + (−0.637 − 0.770i)12-s + (−0.0627 − 0.998i)13-s + (0.968 − 0.248i)14-s + (0.968 + 0.248i)16-s + (0.187 − 0.982i)17-s − 18-s + (0.425 + 0.904i)19-s + (0.425 − 0.904i)21-s + ⋯
L(s)  = 1  + (−0.0627 + 0.998i)2-s + (0.728 + 0.684i)3-s + (−0.992 − 0.125i)4-s + (−0.728 + 0.684i)6-s + (−0.309 − 0.951i)7-s + (0.187 − 0.982i)8-s + (0.0627 + 0.998i)9-s + (−0.637 − 0.770i)12-s + (−0.0627 − 0.998i)13-s + (0.968 − 0.248i)14-s + (0.968 + 0.248i)16-s + (0.187 − 0.982i)17-s − 18-s + (0.425 + 0.904i)19-s + (0.425 − 0.904i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.508 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.508 - 0.860i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1086, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.508 - 0.860i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06918894341 + 0.1212421741i\)
\(L(\frac12)\) \(\approx\) \(-0.06918894341 + 0.1212421741i\)
\(L(1)\) \(\approx\) \(0.8494377510 + 0.4842186737i\)
\(L(1)\) \(\approx\) \(0.8494377510 + 0.4842186737i\)

Euler product

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

−19.904446071257669462764300783756, −19.26439887328883413512073401980, −19.075223612055625847948128544235, −17.985377554959259561392593074058, −17.64746821442000235647293238511, −16.42381195560816869649688051779, −15.33314187690964382060132158449, −14.61738185979814359386298504680, −13.81761204738291324262229700491, −13.14308354218090025003007874044, −12.40937424926501170294027560013, −11.83916408747376216561129591742, −11.040987080777296932363967286805, −9.84865887286101844992157206927, −9.148924648002081564829754650090, −8.71856205800512670416951374726, −7.77196672215578286733476054630, −6.77671871771704666420202494574, −5.78212280634686119144841672406, −4.75359398004526776228752892977, −3.58117939797677292244860803960, −3.01758421651980385981191113352, −1.97657004566867694558891622447, −1.463416366881756838274727438180, −0.026008304676720420438082543215, 1.0835733912410361590521175862, 2.75223011124882256715160916193, 3.62438535778492446907697577709, 4.37376413601377559878245478626, 5.195146416226860311327835989885, 6.109215830264600506124970977707, 7.25150261989334521822507388375, 7.77272026388275501372426167331, 8.512269830035777171720674339366, 9.501893788140582358805843299678, 10.080598601285384275022041032768, 10.658448726285699730805397296169, 12.0770026149044446507275942506, 13.27862313030025644058048213302, 13.59200430094342758857079395534, 14.47061859596160451941706858450, 15.0606372593334713349945600117, 15.8739620377773302115020396859, 16.52300702299720768552638177773, 17.03414611020498277415545529049, 18.073854664311459285552748955206, 18.84233834133928876533461082359, 19.6445265294353277765357954519, 20.52418447470785146718987799720, 20.92971622206545202071316481445

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