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Properties

Label 1-133-133.79-r1-0-0
Degree $1$
Conductor $133$
Sign $-0.401 + 0.915i$
Analytic cond. $14.2928$
Root an. cond. $14.2928$
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  + (−0.766 + 0.642i)2-s + (−0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)6-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.766 − 0.642i)10-s + 11-s − 12-s + (−0.766 − 0.642i)13-s + (0.939 − 0.342i)15-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + (0.5 − 0.866i)18-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.173 − 0.984i)3-s + (0.173 − 0.984i)4-s + (0.173 + 0.984i)5-s + (0.766 + 0.642i)6-s + (0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.766 − 0.642i)10-s + 11-s − 12-s + (−0.766 − 0.642i)13-s + (0.939 − 0.342i)15-s + (−0.939 − 0.342i)16-s + (−0.939 − 0.342i)17-s + (0.5 − 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.401 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(133\)    =    \(7 \cdot 19\)
Sign: $-0.401 + 0.915i$
Analytic conductor: \(14.2928\)
Root analytic conductor: \(14.2928\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{133} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 133,\ (1:\ ),\ -0.401 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3670280131 + 0.5616103302i\)
\(L(\frac12)\) \(\approx\) \(0.3670280131 + 0.5616103302i\)
\(L(1)\) \(\approx\) \(0.6272189927 + 0.1525345671i\)
\(L(1)\) \(\approx\) \(0.6272189927 + 0.1525345671i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.766 + 0.642i)T \)
3 \( 1 + (-0.173 - 0.984i)T \)
5 \( 1 + (0.173 + 0.984i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.766 - 0.642i)T \)
17 \( 1 + (-0.939 - 0.342i)T \)
23 \( 1 + (0.766 + 0.642i)T \)
29 \( 1 + (-0.173 + 0.984i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (-0.766 + 0.642i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (-0.173 + 0.984i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.766 + 0.642i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.173 - 0.984i)T \)
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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

−28.22142844391862909743229917727, −27.10482546085810246936292609642, −26.53141329295409879617166689037, −25.25070107068885743301929031406, −24.33489559247518215854001642436, −22.61940136757086501023092181625, −21.75383036502787118397239597904, −20.93234895956108724956609114131, −20.04987305960891536091175860264, −19.25280580339155976452481027021, −17.52202793186219818852782141957, −16.96254398579220014409072442343, −16.17999654788907904028161879836, −14.86264276528597346497342275041, −13.31017769854478193660703449497, −12.031265178729016184648357275134, −11.27781978628453386906405383251, −9.884933799236530768798739255244, −9.212795283984463588739066165025, −8.34426796859245977310655317159, −6.54019883192114127074773528374, −4.759317757052791881794989080387, −3.884060296960013331122993679495, −2.112215398943973282388753628353, −0.36334410016560820657694399221, 1.378692853472138116198414690851, 2.79933981607399058481214277671, 5.243969189282451493685236146, 6.613539396978121689115612795601, 7.02676355260691673026788906687, 8.28640639750437817253238633597, 9.56542237054476723374557477011, 10.84819564803413604290706488338, 11.7736759810330786035686432866, 13.40485484372019834239173269749, 14.42387682360652709734876424818, 15.2350245875459162306973713281, 16.82070162408345149156054723829, 17.64234896070085138072412424538, 18.3314549779652621820468268884, 19.39936622902966142006619488654, 19.98862091015338423374581291017, 22.082654165221337883308052364076, 22.87106395951144970308558756399, 23.91245242683269408613200507017, 25.00112330308477676458492845026, 25.420884444388038245711032293102, 26.72272651057466474614043422568, 27.446844217528420536029614687074, 28.80454444078585408441573859731

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