Properties

Label 1-407-407.71-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.119 - 0.992i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (−0.719 − 0.694i)2-s + (0.990 − 0.139i)3-s + (0.0348 + 0.999i)4-s + (−0.241 − 0.970i)5-s + (−0.809 − 0.587i)6-s + (0.848 − 0.529i)7-s + (0.669 − 0.743i)8-s + (0.961 − 0.275i)9-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)12-s + (0.559 − 0.829i)13-s + (−0.978 − 0.207i)14-s + (−0.374 − 0.927i)15-s + (−0.997 + 0.0697i)16-s + (0.559 + 0.829i)17-s + (−0.882 − 0.469i)18-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s + (0.990 − 0.139i)3-s + (0.0348 + 0.999i)4-s + (−0.241 − 0.970i)5-s + (−0.809 − 0.587i)6-s + (0.848 − 0.529i)7-s + (0.669 − 0.743i)8-s + (0.961 − 0.275i)9-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)12-s + (0.559 − 0.829i)13-s + (−0.978 − 0.207i)14-s + (−0.374 − 0.927i)15-s + (−0.997 + 0.0697i)16-s + (0.559 + 0.829i)17-s + (−0.882 − 0.469i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.119 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.119 - 0.992i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.119 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9546903016 - 1.076026695i\)
\(L(\frac12)\) \(\approx\) \(0.9546903016 - 1.076026695i\)
\(L(1)\) \(\approx\) \(0.9808557743 - 0.5950926820i\)
\(L(1)\) \(\approx\) \(0.9808557743 - 0.5950926820i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.719 - 0.694i)T \)
3 \( 1 + (0.990 - 0.139i)T \)
5 \( 1 + (-0.241 - 0.970i)T \)
7 \( 1 + (0.848 - 0.529i)T \)
13 \( 1 + (0.559 - 0.829i)T \)
17 \( 1 + (0.559 + 0.829i)T \)
19 \( 1 + (0.990 - 0.139i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.615 - 0.788i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.978 - 0.207i)T \)
53 \( 1 + (-0.241 + 0.970i)T \)
59 \( 1 + (-0.374 - 0.927i)T \)
61 \( 1 + (0.961 + 0.275i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.719 + 0.694i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.997 - 0.0697i)T \)
83 \( 1 + (0.559 + 0.829i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.104 - 0.994i)T \)
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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

−24.59985732865898521437636902911, −24.12246473349849470387701136813, −22.95456789202450665730228539918, −21.98117841343631858598620526519, −20.847830113067358082691917982643, −20.1575366362172716434042293541, −18.9217354169309641122506182285, −18.569725211585751832861579724977, −17.92196949849175731443783491705, −16.411246544082825640729630018565, −15.769660786811205350553216821972, −14.6773404078711911187052111640, −14.45697071803648699987329275453, −13.54180548612895814548571316857, −11.751934267538182511499437828631, −10.96650893675802503342881803446, −9.83629831717551302587002630083, −9.12161390290184232834398901014, −8.05135657430154174322322368116, −7.505391601036746729495520689157, −6.48308715855376018824584662814, −5.22497735837332469878339723490, −3.95848692837728809083406771286, −2.59251307419287709585929079945, −1.60548727262014932733509489880, 1.12389703746639168786174543017, 1.78035448089007656509061047670, 3.38727475125697448309695497545, 4.00728087711158312790329324306, 5.34835109661243783672117451620, 7.36171432825297625232985278748, 7.92240691069511181689964042253, 8.62933848096757575005606523261, 9.517260761283789769322272635567, 10.478400934322500643463134117104, 11.53327561113539525609440965542, 12.56971039009890012466197062022, 13.248077998490563602526469409364, 14.16363777457592180156821475646, 15.43597993951290969525295104527, 16.269104851633964750480738724610, 17.343844486199643588506518747097, 18.048154503255132516162264340633, 19.0609734271338678932082927237, 20.00909192021678788046294397506, 20.397897741006390903496279730017, 21.03674855227523123742332372687, 21.93968322055196354679247479761, 23.455981069692244009193099718850, 24.2637470746086943630419159411

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