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Properties

Label 1-407-407.268-r0-0-0
Degree $1$
Conductor $407$
Sign $0.972 + 0.231i$
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  + (−0.615 − 0.788i)2-s + (0.559 + 0.829i)3-s + (−0.241 + 0.970i)4-s + (0.990 + 0.139i)5-s + (0.309 − 0.951i)6-s + (−0.719 − 0.694i)7-s + (0.913 − 0.406i)8-s + (−0.374 + 0.927i)9-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (0.848 + 0.529i)13-s + (−0.104 + 0.994i)14-s + (0.438 + 0.898i)15-s + (−0.882 − 0.469i)16-s + (0.848 − 0.529i)17-s + (0.961 − 0.275i)18-s + ⋯
L(s)  = 1  + (−0.615 − 0.788i)2-s + (0.559 + 0.829i)3-s + (−0.241 + 0.970i)4-s + (0.990 + 0.139i)5-s + (0.309 − 0.951i)6-s + (−0.719 − 0.694i)7-s + (0.913 − 0.406i)8-s + (−0.374 + 0.927i)9-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (0.848 + 0.529i)13-s + (−0.104 + 0.994i)14-s + (0.438 + 0.898i)15-s + (−0.882 − 0.469i)16-s + (0.848 − 0.529i)17-s + (0.961 − 0.275i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.972 + 0.231i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.972 + 0.231i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.301387921 + 0.1529728762i\)
\(L(\frac12)\) \(\approx\) \(1.301387921 + 0.1529728762i\)
\(L(1)\) \(\approx\) \(1.067109522 + 0.004309906815i\)
\(L(1)\) \(\approx\) \(1.067109522 + 0.004309906815i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.615 - 0.788i)T \)
3 \( 1 + (0.559 + 0.829i)T \)
5 \( 1 + (0.990 + 0.139i)T \)
7 \( 1 + (-0.719 - 0.694i)T \)
13 \( 1 + (0.848 + 0.529i)T \)
17 \( 1 + (0.848 - 0.529i)T \)
19 \( 1 + (0.559 + 0.829i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.997 + 0.0697i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (0.990 - 0.139i)T \)
59 \( 1 + (0.438 + 0.898i)T \)
61 \( 1 + (-0.374 - 0.927i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.615 + 0.788i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.882 + 0.469i)T \)
83 \( 1 + (0.848 - 0.529i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.669 - 0.743i)T \)
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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

−24.5083379496956758962025678808, −23.776932239830448242180068898963, −22.78810782211815866117283899313, −21.77578440689557605340056723128, −20.56779794092037076413287521878, −19.717132350578224478430197774866, −18.80750038555206580391819680734, −18.1776474955106143526430731013, −17.51874617241163085835816156346, −16.47999761899368938848649169489, −15.47515175728220343540435788025, −14.66423563592495053410175836513, −13.565074439863729142397706609665, −13.18872247868533497924264947384, −11.948178708217839711159372945982, −10.45085306666748006388330028583, −9.47099279545579636418999122246, −8.88708311832571990051176366727, −7.947591070714285549593489277241, −6.861662001311667001574887687068, −5.99331331489401144287330364590, −5.42940933226596323383358961802, −3.37066171591240165950149541023, −2.124737336171427481245999855953, −1.0516666058918923977715838148, 1.329116670408283399851168523723, 2.62186244772191719307734206179, 3.47434537540065740018245730374, 4.41421007603205598841457355880, 5.8722061030044587192428491281, 7.187732102337615342225260849996, 8.3625007231934662232291816143, 9.27366830304967992124563364871, 10.06983379293200737715885088942, 10.40476789886315946281717083245, 11.614560714630884475332759136113, 12.8801773796843882005150193328, 13.81800644051895026805293004948, 14.24268114606387186203945309978, 16.00255915766609163551343324390, 16.47685094412249168092887598759, 17.332807714220314844903037910829, 18.499619781486090587765932306103, 19.13897001293269297693138945895, 20.29753931894284255746482366293, 20.76786551075172906750412449474, 21.4492563650358246164040433413, 22.42372191643740104044602818430, 23.00776934677652719168973803728, 24.83228804102207680694508238523

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