Properties

Label 1-407-407.294-r0-0-0
Degree $1$
Conductor $407$
Sign $0.936 + 0.350i$
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  + (0.469 − 0.882i)2-s + (0.719 + 0.694i)3-s + (−0.559 − 0.829i)4-s + (−0.529 + 0.848i)5-s + (0.951 − 0.309i)6-s + (0.997 − 0.0697i)7-s + (−0.994 + 0.104i)8-s + (0.0348 + 0.999i)9-s + (0.5 + 0.866i)10-s + (0.173 − 0.984i)12-s + (0.788 + 0.615i)13-s + (0.406 − 0.913i)14-s + (−0.970 + 0.241i)15-s + (−0.374 + 0.927i)16-s + (−0.788 + 0.615i)17-s + (0.898 + 0.438i)18-s + ⋯
L(s)  = 1  + (0.469 − 0.882i)2-s + (0.719 + 0.694i)3-s + (−0.559 − 0.829i)4-s + (−0.529 + 0.848i)5-s + (0.951 − 0.309i)6-s + (0.997 − 0.0697i)7-s + (−0.994 + 0.104i)8-s + (0.0348 + 0.999i)9-s + (0.5 + 0.866i)10-s + (0.173 − 0.984i)12-s + (0.788 + 0.615i)13-s + (0.406 − 0.913i)14-s + (−0.970 + 0.241i)15-s + (−0.374 + 0.927i)16-s + (−0.788 + 0.615i)17-s + (0.898 + 0.438i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.936 + 0.350i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (294, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.936 + 0.350i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.881419418 + 0.3404532682i\)
\(L(\frac12)\) \(\approx\) \(1.881419418 + 0.3404532682i\)
\(L(1)\) \(\approx\) \(1.528589695 - 0.03908114669i\)
\(L(1)\) \(\approx\) \(1.528589695 - 0.03908114669i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.469 - 0.882i)T \)
3 \( 1 + (0.719 + 0.694i)T \)
5 \( 1 + (-0.529 + 0.848i)T \)
7 \( 1 + (0.997 - 0.0697i)T \)
13 \( 1 + (0.788 + 0.615i)T \)
17 \( 1 + (-0.788 + 0.615i)T \)
19 \( 1 + (-0.694 + 0.719i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.406 + 0.913i)T \)
31 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.961 - 0.275i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (0.848 - 0.529i)T \)
59 \( 1 + (0.970 - 0.241i)T \)
61 \( 1 + (0.999 + 0.0348i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.882 + 0.469i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.927 + 0.374i)T \)
83 \( 1 + (0.615 + 0.788i)T \)
89 \( 1 + (0.642 - 0.766i)T \)
97 \( 1 + (-0.207 - 0.978i)T \)
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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

−24.42711355453571950074932636479, −23.54633726203215864223168529808, −23.07928762185327013731210383585, −21.55730585941644919836759535929, −20.813147785400547675710515635816, −20.10587957665517618623764736589, −18.99527541190967066918495299508, −17.85681304836918104971858323091, −17.46246554595660029862346681019, −16.15129695319968201965759895160, −15.30607002458661497448938683660, −14.71180852392856901243555964802, −13.45901878624865664695431198383, −13.14820458132939757100817029778, −12.060627600701082718745245426228, −11.1917756063311245471916251483, −9.11731957551310763397932175168, −8.650278421756463260068401249, −7.82790397238857956161525797118, −7.08828190595604082593347250435, −5.82071018124477072867200863710, −4.74035622704366581047480852617, −3.87558209688957169626712714389, −2.5545986233431808497328721530, −0.95798777425971905418146359400, 1.72396820678603423580482404409, 2.65010293143893754879486718966, 3.91144817657146650235490676772, 4.25727806221293416651144285349, 5.588503084809432482362698620563, 6.9914121753315482700348587137, 8.38737813806533819117493929949, 8.95337028871836647569520007202, 10.44772295710197728669179880684, 10.81000877441835979438702466569, 11.58878323844515862826458204603, 12.88836636632108922984571802163, 13.95035646004899767079876214173, 14.64010445999722169592414835052, 15.09409497628970827172811204819, 16.20946977605304575169126631565, 17.65370409169268471829567596752, 18.700452058825045271245036129970, 19.23006819331320325677491180124, 20.25951071212220623942558777789, 20.9049395796651528348819334380, 21.66966051326270547581097542240, 22.36499951176311936891646756654, 23.411012609304868638495086773800, 24.05899926565087056496611262956

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