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Properties

Label 1-1375-1375.1129-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.498 - 0.866i$
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  + (−0.187 − 0.982i)2-s + (−0.0627 − 0.998i)3-s + (−0.929 + 0.368i)4-s + (−0.968 + 0.248i)6-s + (−0.809 − 0.587i)7-s + (0.535 + 0.844i)8-s + (−0.992 + 0.125i)9-s + (0.425 + 0.904i)12-s + (−0.992 + 0.125i)13-s + (−0.425 + 0.904i)14-s + (0.728 − 0.684i)16-s + (0.0627 − 0.998i)17-s + (0.309 + 0.951i)18-s + (−0.968 + 0.248i)19-s + (−0.535 + 0.844i)21-s + ⋯
L(s)  = 1  + (−0.187 − 0.982i)2-s + (−0.0627 − 0.998i)3-s + (−0.929 + 0.368i)4-s + (−0.968 + 0.248i)6-s + (−0.809 − 0.587i)7-s + (0.535 + 0.844i)8-s + (−0.992 + 0.125i)9-s + (0.425 + 0.904i)12-s + (−0.992 + 0.125i)13-s + (−0.425 + 0.904i)14-s + (0.728 − 0.684i)16-s + (0.0627 − 0.998i)17-s + (0.309 + 0.951i)18-s + (−0.968 + 0.248i)19-s + (−0.535 + 0.844i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.498 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.498 - 0.866i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.498 - 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3323886805 - 0.5746016027i\)
\(L(\frac12)\) \(\approx\) \(0.3323886805 - 0.5746016027i\)
\(L(1)\) \(\approx\) \(0.4353705708 - 0.4514464520i\)
\(L(1)\) \(\approx\) \(0.4353705708 - 0.4514464520i\)

Euler product

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

−21.288281833759818542396462559192, −19.824377403801943954384070331190, −19.47303342018347782007259243443, −18.59422470297517175190504563692, −17.410096322529942933189786658957, −17.1429772798446521894764235697, −16.23953522005838487302283713896, −15.58264129202623331759539122160, −15.02542801757079646165781291969, −14.442350775682759954238334143430, −13.36946998217357538500343863478, −12.60555616868865546244537204865, −11.638853442556537195802930460631, −10.45024773170155817721410068538, −9.831355319618678734262057062761, −9.303421853814700889936639761109, −8.38723481419131123905921439076, −7.70820714538825975983527864998, −6.236905256863992924298691112358, −6.128602566564052910799850193197, −4.9669472396347628681445768566, −4.27763242097043617644370788888, −3.35271936915778698769339803411, −2.20516551997843349940332666912, −0.34240025186503109689052843425, 0.39825190644533361461172444540, 1.34201353363152444214203756275, 2.48959894298940699330660370465, 2.98256613097454086739880094278, 4.195928246870570421809777427792, 5.06382332133028638669565609986, 6.288773040578574792779204177138, 7.05388647036381389608897304252, 7.89392162997905057225093080251, 8.713083168934066579238289480631, 9.66595402812184181524413810490, 10.321100961453505504316383436038, 11.2293055857689242357044750374, 12.08243687318270552094160404832, 12.6248609420903770875848165783, 13.234166358463968413379205867028, 14.11511491888357100090012397049, 14.5831449707370184319463336893, 16.24742291050487295188916798583, 16.76748736443195264121770710567, 17.677816829208485378674588264428, 18.20943080060228940477185338016, 19.09676884995248143354376175935, 19.64127158282818404007388540745, 20.06932942629173268574499265967

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