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Properties

Label 1-429-429.296-r0-0-0
Degree $1$
Conductor $429$
Sign $-0.872 - 0.488i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(429\)    =    \(3 \cdot 11 \cdot 13\)
Sign: $-0.872 - 0.488i$
Analytic conductor: \(1.99226\)
Root analytic conductor: \(1.99226\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{429} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 429,\ (0:\ ),\ -0.872 - 0.488i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3931198616 - 1.505694673i\)
\(L(\frac12)\) \(\approx\) \(0.3931198616 - 1.505694673i\)
\(L(1)\) \(\approx\) \(0.9629586238 - 0.8932112614i\)
\(L(1)\) \(\approx\) \(0.9629586238 - 0.8932112614i\)

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

−24.64683701500452470172030994628, −23.73705342846628543494160839955, −22.755266479705319263862664999520, −21.90766741916810174969609971291, −21.498622313628409957269675235628, −20.509558043725491446326787687246, −19.045212738819254417458524451824, −18.31301857588570099127744785112, −17.31757081601888247995542941714, −16.760217278655215951608139564624, −15.631788285819639634167994222790, −14.94497455115609697113378600622, −14.05916725209751045939387716952, −13.01936831673380744807096595734, −12.640518413755210168174926595258, −11.34427006725379439926468630096, −9.92956786622452603200426961135, −9.12763961486102055102075674810, −8.28835237877712871861485831618, −7.02244285217499715945378537143, −5.95850778793436739649637650987, −5.66253467455857101546334228493, −4.30018935784243324609476820146, −3.08695461817675601533743734153, −1.945467547190978266640464936637, 0.7403195340873897799861561374, 2.10894068900305779762956786114, 3.02624773002616878423564820589, 4.25723226153958580065185243036, 5.17465424088727617715525397074, 6.28164243688595016760831043727, 7.14999151123487455934126614112, 9.0162511083985521283149035165, 9.481496057972852634366856310757, 10.67302158928247620555770763250, 11.02883404387241758340308962099, 12.617286917665325412741362364903, 13.075590253509907957044502749046, 13.968780172931315587931991225945, 14.5978816907016291172163369719, 15.90972903209002368534337892264, 16.98204276049609335945564429659, 17.87435241828390927122989954856, 18.67929188072437601983367798776, 19.75419606384984529378004689561, 20.404856167745619936367852021690, 21.17066494225891668714057684727, 22.12313363955060727521670705591, 22.64516508924701937058666474417, 23.65282324062001964513089819409

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