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Properties

Label 1-411-411.410-r1-0-0
Degree $1$
Conductor $411$
Sign $1$
Analytic cond. $44.1680$
Root an. cond. $44.1680$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(411\)    =    \(3 \cdot 137\)
Sign: $1$
Analytic conductor: \(44.1680\)
Root analytic conductor: \(44.1680\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{411} (410, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 411,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.626467650\)
\(L(\frac12)\) \(\approx\) \(1.626467650\)
\(L(1)\) \(\approx\) \(0.9297800258\)
\(L(1)\) \(\approx\) \(0.9297800258\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
137 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 + T \)
59 \( 1 - T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + T \)
97 \( 1 - T \)
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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

−24.499101606547688483127997506376, −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