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Properties

Label 1-1375-1375.1031-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: $\chi_{1375} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.498 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.101977211 - 1.904992284i\)
\(L(\frac12)\) \(\approx\) \(-1.101977211 - 1.904992284i\)
\(L(1)\) \(\approx\) \(0.5667291298 - 1.052911012i\)
\(L(1)\) \(\approx\) \(0.5667291298 - 1.052911012i\)

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

−21.33486987570054026065276088307, −20.826637802295905637751635818967, −19.568069244042154017226246786681, −18.77918405309963577710565873927, −17.75393600340944681648051282391, −17.3222961226881655547185250201, −16.50487181557626896982386141623, −15.494476064325335840152845444026, −15.40301381707955584330046409546, −14.42638946258240665629098421582, −13.92280167419991320826547557950, −12.83157864213080512565159980313, −12.01208202814400793747516162846, −10.890938826149141029973757528340, −10.37297528872538488363513801551, −9.075168861285082913117717164387, −8.661187269504575873205380581293, −8.08789078972658315812129305858, −6.87097822951866768744425581430, −5.831152541515879205434361091252, −5.39730266860437922021249437211, −4.38523613076487643574860512087, −3.80272655343182231103369829571, −2.72190064238219650424357582660, −1.2203772030514709302945071900, 0.47296166506824433985677156711, 1.04538970250774117602113382874, 2.0924368800893435374590517638, 2.78764951200179755597415810476, 3.97836509494941369271614251132, 4.73202011943133072882144782423, 5.77831364139976142116587790237, 6.66695759364979865249636329434, 7.68615411799919940060999144525, 8.49976277951994355195917663597, 9.08965884244899841619130154328, 10.34939572452068473360455600189, 11.09598670060705533553590753383, 11.58266323099267260784068381910, 12.45414887197763597234891881593, 13.26855151809122574678245177203, 13.79753353069888533025079928083, 14.400417903903811692923190162975, 15.31490236462291544078437724779, 16.671639231577134645027587404402, 17.5115078276151764124875057528, 17.9765717071533259161456304259, 18.82687224902200589411553461796, 19.29637347620417148723419056031, 20.24399742114392972176220491807

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