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Properties

Label 1-1375-1375.19-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.670 + 0.742i$
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  + (−0.992 + 0.125i)2-s + (−0.535 − 0.844i)3-s + (0.968 − 0.248i)4-s + (0.637 + 0.770i)6-s + (−0.809 + 0.587i)7-s + (−0.929 + 0.368i)8-s + (−0.425 + 0.904i)9-s + (−0.728 − 0.684i)12-s + (−0.425 + 0.904i)13-s + (0.728 − 0.684i)14-s + (0.876 − 0.481i)16-s + (0.535 − 0.844i)17-s + (0.309 − 0.951i)18-s + (0.637 + 0.770i)19-s + (0.929 + 0.368i)21-s + ⋯
L(s)  = 1  + (−0.992 + 0.125i)2-s + (−0.535 − 0.844i)3-s + (0.968 − 0.248i)4-s + (0.637 + 0.770i)6-s + (−0.809 + 0.587i)7-s + (−0.929 + 0.368i)8-s + (−0.425 + 0.904i)9-s + (−0.728 − 0.684i)12-s + (−0.425 + 0.904i)13-s + (0.728 − 0.684i)14-s + (0.876 − 0.481i)16-s + (0.535 − 0.844i)17-s + (0.309 − 0.951i)18-s + (0.637 + 0.770i)19-s + (0.929 + 0.368i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.670 + 0.742i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.670 + 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4756962899 + 0.2113161888i\)
\(L(\frac12)\) \(\approx\) \(0.4756962899 + 0.2113161888i\)
\(L(1)\) \(\approx\) \(0.4895975063 - 0.04089140381i\)
\(L(1)\) \(\approx\) \(0.4895975063 - 0.04089140381i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.992 + 0.125i)T \)
3 \( 1 + (-0.535 - 0.844i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.425 + 0.904i)T \)
17 \( 1 + (0.535 - 0.844i)T \)
19 \( 1 + (0.637 + 0.770i)T \)
23 \( 1 + (-0.876 - 0.481i)T \)
29 \( 1 + (-0.0627 + 0.998i)T \)
31 \( 1 + (-0.929 + 0.368i)T \)
37 \( 1 + (0.425 - 0.904i)T \)
41 \( 1 + (0.425 - 0.904i)T \)
43 \( 1 + (-0.809 - 0.587i)T \)
47 \( 1 + (0.637 - 0.770i)T \)
53 \( 1 + (0.637 - 0.770i)T \)
59 \( 1 + (-0.187 - 0.982i)T \)
61 \( 1 + (0.187 - 0.982i)T \)
67 \( 1 + (-0.535 + 0.844i)T \)
71 \( 1 + (0.535 + 0.844i)T \)
73 \( 1 + (-0.992 + 0.125i)T \)
79 \( 1 + (0.929 + 0.368i)T \)
83 \( 1 + (-0.929 + 0.368i)T \)
89 \( 1 + (0.728 - 0.684i)T \)
97 \( 1 + (-0.968 + 0.248i)T \)
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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

−20.232866545027675576159557167619, −19.972209919454217953741533789594, −19.1181286890356615314151153331, −18.03216755514598439847252369773, −17.53039166593568498580717076694, −16.67366457544440960073202798219, −16.34253604421717115891777749988, −15.327792713381845101791614044999, −14.97949874972984623670695658729, −13.573803072783165023995589918140, −12.61544307823367260223734117892, −11.85446133106438927647206278006, −11.03717150584199658105709193111, −10.260807740255491220074720286880, −9.81666792697289947271519927295, −9.15906509795010391812439490828, −8.00315815754740608836783740287, −7.32465263010709505691242353692, −6.19991426794947363015327920469, −5.72727107215124712209260487624, −4.381062327300762016602322297248, −3.43468876000980454070733480790, −2.75118811544903558763509139315, −1.18498838596694300495074291740, −0.275527568300303324340055055030, 0.57600871519417308268932573867, 1.77173667776157651071992313442, 2.4366926804364570628783063947, 3.552731472635986032570156283948, 5.29020856134672644402358926337, 5.80301674746813809380447502165, 6.85627619293374930909916313335, 7.18531441746547968972632659207, 8.20180482835525302298897581081, 9.059098750644905630349170916511, 9.79052528766765453375285089930, 10.61535397150873629400562057775, 11.64199937694687234018408481612, 12.11496578544451575013304849311, 12.740586197330123753391419671386, 14.01415646373377610055439554027, 14.59402942080575744856490513781, 15.99636234697845527381661009184, 16.27116496436388332184483838566, 16.94454888001207441930002695262, 17.96592043595787602546347528378, 18.53328657629359665781040480606, 18.9103303368324135275334883678, 19.795107275638137524826230147492, 20.38272344480280573704308869750

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