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Properties

Label 1-407-407.229-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.540 - 0.841i$
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  + (−0.997 + 0.0697i)2-s + (0.848 − 0.529i)3-s + (0.990 − 0.139i)4-s + (0.559 − 0.829i)5-s + (−0.809 + 0.587i)6-s + (−0.615 − 0.788i)7-s + (−0.978 + 0.207i)8-s + (0.438 − 0.898i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (−0.719 + 0.694i)13-s + (0.669 + 0.743i)14-s + (0.0348 − 0.999i)15-s + (0.961 − 0.275i)16-s + (−0.719 − 0.694i)17-s + (−0.374 + 0.927i)18-s + ⋯
L(s)  = 1  + (−0.997 + 0.0697i)2-s + (0.848 − 0.529i)3-s + (0.990 − 0.139i)4-s + (0.559 − 0.829i)5-s + (−0.809 + 0.587i)6-s + (−0.615 − 0.788i)7-s + (−0.978 + 0.207i)8-s + (0.438 − 0.898i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (−0.719 + 0.694i)13-s + (0.669 + 0.743i)14-s + (0.0348 − 0.999i)15-s + (0.961 − 0.275i)16-s + (−0.719 − 0.694i)17-s + (−0.374 + 0.927i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.540 - 0.841i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.540 - 0.841i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4832740280 - 0.8843075116i\)
\(L(\frac12)\) \(\approx\) \(0.4832740280 - 0.8843075116i\)
\(L(1)\) \(\approx\) \(0.7801477446 - 0.4318742845i\)
\(L(1)\) \(\approx\) \(0.7801477446 - 0.4318742845i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.997 + 0.0697i)T \)
3 \( 1 + (0.848 - 0.529i)T \)
5 \( 1 + (0.559 - 0.829i)T \)
7 \( 1 + (-0.615 - 0.788i)T \)
13 \( 1 + (-0.719 + 0.694i)T \)
17 \( 1 + (-0.719 - 0.694i)T \)
19 \( 1 + (0.848 - 0.529i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.882 - 0.469i)T \)
43 \( 1 + T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (0.559 + 0.829i)T \)
59 \( 1 + (0.0348 - 0.999i)T \)
61 \( 1 + (0.438 + 0.898i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.997 - 0.0697i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.961 + 0.275i)T \)
83 \( 1 + (-0.719 - 0.694i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.913 - 0.406i)T \)
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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

−25.18929921488622416010976332287, −24.20778405935929170940406561080, −22.38012223062060166982386149647, −21.96984819660282474728415304814, −21.07402897414609712280567178305, −20.00745327868171871579092083851, −19.47681906556098731778216556675, −18.47866811746746164227048944708, −17.944457081267318807596470373287, −16.70070606900859152982885456070, −15.812558061797896060294588256114, −15.04997550881058524891985843714, −14.372512145736234523538517892594, −13.04656549193261748154156048326, −12.03956984589147694120709248735, −10.609464020152885271028920404238, −10.18658274553111417897697272999, −9.29762662285599188795325514154, −8.52352974141101042508489594286, −7.4613603705142492979867549016, −6.48019169462181266395827386356, −5.38509065711966406423531374130, −3.50286127254151402020839906389, −2.71509863253402432245024904652, −1.93121074592152929068934242277, 0.69648428273769796121669628309, 1.86250133428509910583217548453, 2.83920252231617869222416879759, 4.27755073275738236292139095378, 5.87378911509642522098050441852, 7.01901348477506041672646377405, 7.54720492318062191946533240656, 8.79824743928279493388092773643, 9.45427855064692611978039845511, 10.00367067213753886897601376282, 11.523880614536404932800251315403, 12.4479619243844928346763857158, 13.51388792546499725021649418912, 14.09995000538176697807146363830, 15.5026261766091164468032935651, 16.22657741627421248953756387912, 17.236751306360936711582241893185, 17.832512445385384406678531060136, 18.92878039866464097696935870026, 19.76828159432683680784380050169, 20.21476935905606179460570749338, 20.993412912231981295197909960524, 22.10482784253994797639958667334, 23.77289564567706298598946440571, 24.18674535151375255112251161911

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