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Properties

Label 1-429-429.230-r0-0-0
Degree $1$
Conductor $429$
Sign $-0.711 - 0.702i$
Analytic cond. $1.99226$
Root an. cond. $1.99226$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)7-s + 8-s + (0.5 + 0.866i)10-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (0.5 + 0.866i)23-s + 25-s + (0.5 + 0.866i)28-s + (−0.5 − 0.866i)29-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)4-s − 5-s + (0.5 − 0.866i)7-s + 8-s + (0.5 + 0.866i)10-s − 14-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 − 0.866i)20-s + (0.5 + 0.866i)23-s + 25-s + (0.5 + 0.866i)28-s + (−0.5 − 0.866i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.711 - 0.702i$
Analytic conductor: \(1.99226\)
Root analytic conductor: \(1.99226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (230, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 429,\ (0:\ ),\ -0.711 - 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2544972027 - 0.6200253707i\)
\(L(\frac12)\) \(\approx\) \(0.2544972027 - 0.6200253707i\)
\(L(1)\) \(\approx\) \(0.5792598389 - 0.3579130377i\)
\(L(1)\) \(\approx\) \(0.5792598389 - 0.3579130377i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 - T \)
7 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−24.5616277677215903172610979924, −23.85310596438967576238422889074, −22.808812917729888357532144292942, −22.32981037500495914285411486815, −20.86769908870117055933038263943, −20.01770114146313791514087880107, −18.93722012518193898066488569904, −18.48280454149171911971819267542, −17.59127302463760240582084204699, −16.3862283510239370147260316554, −15.888652591882838125051359191844, −14.9267191167061062081680164162, −14.41257612900320247095021597779, −13.10040756258885276788638179661, −11.92426023846112218031211116960, −11.1743057713018660244696845084, −9.9985921110355713088461279953, −8.86373655409012969836727513553, −8.26226506280050435948331523422, −7.35998168251310781154381211752, −6.39123019399059792409905261951, −5.178535098589507585097409574145, −4.46058282762186312925836538959, −2.91297519926519385415547010810, −1.28189000213056364830781627368, 0.521182431283176396007387577615, 1.830467306537498262993759039412, 3.292080918022602822907893277744, 4.07249364720945923801294300297, 4.97096189812365330052341984977, 6.90772458594519128987147877, 7.71259647818075473458431684497, 8.475590102316394563886376812263, 9.53820347418048357442500061446, 10.71553701710136449214336466583, 11.21667757551144774113489025527, 12.06692842787877246631203734418, 13.11336089758016305295073218706, 13.90787170135425091525421560239, 15.17606952436229303483831380438, 16.07750978580977746988992783773, 17.2538724676507676002194399464, 17.62885597658416817912930235222, 18.99185569278508760328751961754, 19.476272973524537595081738156370, 20.30334148186191645149288067705, 20.97869746345346465096810675543, 22.03186074847656566019436268142, 22.90657056630670417287066230407, 23.69091746347823899679284177695

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