Properties

Degree 1
Conductor $ 3 \cdot 137 $
Sign $1$
Motivic weight 0
Primitive yes
Self-dual yes
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 11-s − 13-s − 14-s + 16-s − 17-s + 19-s + 20-s + 22-s + 23-s + 25-s + 26-s + 28-s + 29-s − 31-s − 32-s + 34-s + 35-s + 37-s − 38-s − 40-s + 41-s + ⋯
L(s,χ)  = 1  − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s − 11-s − 13-s − 14-s + 16-s − 17-s + 19-s + 20-s + 22-s + 23-s + 25-s + 26-s + 28-s + 29-s − 31-s − 32-s + 34-s + 35-s + 37-s − 38-s − 40-s + 41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 411 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 411 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(411\)    =    \(3 \cdot 137\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(0\)
character  :  $\chi_{411} (410, \cdot )$
Sato-Tate  :  $\mu(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((1,\ 411,\ (1:\ ),\ 1)\)
\(L(\chi,\frac{1}{2})\)  \(\approx\)  \(1.626467650\)
\(L(\frac12,\chi)\)  \(\approx\)  \(1.626467650\)
\(L(\chi,1)\)  \(\approx\)  \(0.9297800258\)
\(L(1,\chi)\)  \(\approx\)  \(0.9297800258\)

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−24.499101606547688483127997506376, −23.49594462544514785994409965374, −21.97187053423888882291495938026, −21.349974622299469165585684232648, −20.53237553474074335291959435686, −19.80608556280237026677086296678, −18.49349075719317787211220147705, −17.97716115404585280618116997501, −17.33366546494640101920810959704, −16.45934259289122820484133581544, −15.340412916882108286169453842723, −14.55363590507511571587946572115, −13.46079757362801050840175668096, −12.375906420822041777329650365414, −11.226312386498288902401453224287, −10.53593875442848032545929262263, −9.60587821067805894594406266452, −8.78489019475498731965836300710, −7.72443506001733004113471737618, −6.93584810436780098614920595491, −5.63406437257690271759422306837, −4.840081861849494663833741353535, −2.77328403999234838611661725137, −2.078118301775149456746025867704, −0.83818836870164467450936180208, 0.83818836870164467450936180208, 2.078118301775149456746025867704, 2.77328403999234838611661725137, 4.840081861849494663833741353535, 5.63406437257690271759422306837, 6.93584810436780098614920595491, 7.72443506001733004113471737618, 8.78489019475498731965836300710, 9.60587821067805894594406266452, 10.53593875442848032545929262263, 11.226312386498288902401453224287, 12.375906420822041777329650365414, 13.46079757362801050840175668096, 14.55363590507511571587946572115, 15.340412916882108286169453842723, 16.45934259289122820484133581544, 17.33366546494640101920810959704, 17.97716115404585280618116997501, 18.49349075719317787211220147705, 19.80608556280237026677086296678, 20.53237553474074335291959435686, 21.349974622299469165585684232648, 21.97187053423888882291495938026, 23.49594462544514785994409965374, 24.499101606547688483127997506376

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