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Properties

Label 1-1375-1375.1103-r1-0-0
Degree $1$
Conductor $1375$
Sign $0.703 - 0.710i$
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  + (0.904 + 0.425i)2-s + (−0.248 + 0.968i)3-s + (0.637 + 0.770i)4-s + (−0.637 + 0.770i)6-s + (−0.951 − 0.309i)7-s + (0.248 + 0.968i)8-s + (−0.876 − 0.481i)9-s + (−0.904 + 0.425i)12-s + (−0.481 + 0.876i)13-s + (−0.728 − 0.684i)14-s + (−0.187 + 0.982i)16-s + (0.844 − 0.535i)17-s + (−0.587 − 0.809i)18-s + (−0.0627 + 0.998i)19-s + (0.535 − 0.844i)21-s + ⋯
L(s)  = 1  + (0.904 + 0.425i)2-s + (−0.248 + 0.968i)3-s + (0.637 + 0.770i)4-s + (−0.637 + 0.770i)6-s + (−0.951 − 0.309i)7-s + (0.248 + 0.968i)8-s + (−0.876 − 0.481i)9-s + (−0.904 + 0.425i)12-s + (−0.481 + 0.876i)13-s + (−0.728 − 0.684i)14-s + (−0.187 + 0.982i)16-s + (0.844 − 0.535i)17-s + (−0.587 − 0.809i)18-s + (−0.0627 + 0.998i)19-s + (0.535 − 0.844i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (1103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ 0.703 - 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08764874862 - 0.03655839293i\)
\(L(\frac12)\) \(\approx\) \(0.08764874862 - 0.03655839293i\)
\(L(1)\) \(\approx\) \(0.9282016494 + 0.7550186451i\)
\(L(1)\) \(\approx\) \(0.9282016494 + 0.7550186451i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.904 + 0.425i)T \)
3 \( 1 + (-0.248 + 0.968i)T \)
7 \( 1 + (-0.951 - 0.309i)T \)
13 \( 1 + (-0.481 + 0.876i)T \)
17 \( 1 + (0.844 - 0.535i)T \)
19 \( 1 + (-0.0627 + 0.998i)T \)
23 \( 1 + (-0.684 + 0.728i)T \)
29 \( 1 + (-0.535 + 0.844i)T \)
31 \( 1 + (0.0627 - 0.998i)T \)
37 \( 1 + (-0.904 + 0.425i)T \)
41 \( 1 + (-0.187 + 0.982i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (-0.844 - 0.535i)T \)
53 \( 1 + (0.770 - 0.637i)T \)
59 \( 1 + (0.187 - 0.982i)T \)
61 \( 1 + (-0.425 + 0.904i)T \)
67 \( 1 + (-0.998 - 0.0627i)T \)
71 \( 1 + (-0.637 - 0.770i)T \)
73 \( 1 + (0.481 + 0.876i)T \)
79 \( 1 + (0.929 - 0.368i)T \)
83 \( 1 + (0.368 - 0.929i)T \)
89 \( 1 + (0.425 - 0.904i)T \)
97 \( 1 + (0.248 - 0.968i)T \)
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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

−20.74789159948863716226903366249, −19.686296773279387199079264790964, −19.55097093458811489340437030467, −18.691482833752163891752967886274, −17.88386104240206749748879987905, −16.92495422396719765642821341195, −16.10514846541548172077182507808, −15.2588220645515631395292554571, −14.49277119460971411217448073577, −13.59496759487979605359476596530, −13.01559032376495659541064669363, −12.30446241878944034729022662652, −11.9579201401356848513693720905, −10.77074087497065752970668694374, −10.1967540949567030331646819737, −9.1374109331240635754446518153, −7.98390151451186843177836808926, −7.09839656961461022458304804445, −6.349700761168756325187067514457, −5.69222939914050029739511331252, −4.92978613994851925947770173771, −3.60877119860912056876203562080, −2.83230330251497061163625505836, −2.099946001613005702815298126208, −0.90687099626370957148865101633, 0.014572505734303752288902307429, 1.88500019410341632712664146137, 3.2242404197995147436443640639, 3.61533594399446983956870806713, 4.527557529790995607453089016039, 5.389505141510817568277381187688, 6.094954981762385371585900883665, 6.92810071691286984176127471824, 7.80828142081992087982392837570, 8.920530763327297815294538009493, 9.82674303603945204341818981415, 10.37936767001822062469731774203, 11.66742417006105288131798804498, 11.91767948201839047760202314934, 12.98261069128223990843503126630, 13.85828401339369669497081766668, 14.49187483490598042675392136730, 15.22344127637939786935241840999, 16.06702487411513227092954758715, 16.65796556689155975817229001878, 16.90406178663607311920203214465, 18.1592442722304374421755768239, 19.25847994950599055249185872693, 20.09527772291372891635487434089, 20.77419678805489102258620561322

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