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Properties

Label 1-407-407.108-r0-0-0
Degree $1$
Conductor $407$
Sign $0.565 - 0.825i$
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  + (0.438 − 0.898i)2-s + (−0.882 − 0.469i)3-s + (−0.615 − 0.788i)4-s + (−0.997 − 0.0697i)5-s + (−0.809 + 0.587i)6-s + (−0.374 + 0.927i)7-s + (−0.978 + 0.207i)8-s + (0.559 + 0.829i)9-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)12-s + (0.961 + 0.275i)13-s + (0.669 + 0.743i)14-s + (0.848 + 0.529i)15-s + (−0.241 + 0.970i)16-s + (0.961 − 0.275i)17-s + (0.990 − 0.139i)18-s + ⋯
L(s)  = 1  + (0.438 − 0.898i)2-s + (−0.882 − 0.469i)3-s + (−0.615 − 0.788i)4-s + (−0.997 − 0.0697i)5-s + (−0.809 + 0.587i)6-s + (−0.374 + 0.927i)7-s + (−0.978 + 0.207i)8-s + (0.559 + 0.829i)9-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)12-s + (0.961 + 0.275i)13-s + (0.669 + 0.743i)14-s + (0.848 + 0.529i)15-s + (−0.241 + 0.970i)16-s + (0.961 − 0.275i)17-s + (0.990 − 0.139i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.565 - 0.825i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.565 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7083983673 - 0.3734574412i\)
\(L(\frac12)\) \(\approx\) \(0.7083983673 - 0.3734574412i\)
\(L(1)\) \(\approx\) \(0.6854594529 - 0.3677466754i\)
\(L(1)\) \(\approx\) \(0.6854594529 - 0.3677466754i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.438 - 0.898i)T \)
3 \( 1 + (-0.882 - 0.469i)T \)
5 \( 1 + (-0.997 - 0.0697i)T \)
7 \( 1 + (-0.374 + 0.927i)T \)
13 \( 1 + (0.961 + 0.275i)T \)
17 \( 1 + (0.961 - 0.275i)T \)
19 \( 1 + (-0.882 - 0.469i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.0348 + 0.999i)T \)
43 \( 1 + T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (-0.997 + 0.0697i)T \)
59 \( 1 + (0.848 + 0.529i)T \)
61 \( 1 + (0.559 - 0.829i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.438 + 0.898i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.241 - 0.970i)T \)
83 \( 1 + (0.961 - 0.275i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
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

−23.8740512875469735084340269612, −23.623753832002034469399355927745, −22.90768468026779177290936791316, −22.30759113399112025207352532991, −21.146954733546402022535561942599, −20.39229038261426564434875411698, −19.03193844623189930910825413663, −18.13130341913873163733641122275, −17.10934711751235810499802732602, −16.32806965891504045668871044253, −15.962858598878636878353689694418, −14.9196508335359036151098818377, −14.07666687687591116328027216893, −12.69886697657803519070160969936, −12.28348381709073160987961504902, −10.95621765438954579058246038473, −10.28464359557969364846003172192, −8.83678216037461878684939845282, −7.85227973871325291131479995005, −6.87360308613819287263531647651, −6.11792312891796117998935493850, −4.97256738341603733694124407463, −3.90594399307351602157439860496, −3.54324911169818287705858424618, −0.66748066347743662669688641120, 0.89998178750916845093266774087, 2.25833546831767997008967919778, 3.53695772108901734867826795135, 4.55117087739105157712980495158, 5.647169565776181187096175629450, 6.39216274633059728014453074860, 7.80907890654209541688648648291, 8.89152939152073236381326862885, 10.04029294309087983899656324143, 11.25945094028170963833178774705, 11.58091027867081810281517251635, 12.52414008972129689866909942598, 13.07587564416049903875333290779, 14.314159209443215451804924048790, 15.51895526059256333016685966981, 16.083837640698067855862365656631, 17.41507699795454671137530036624, 18.5191012478821809382712852315, 18.93267720555096513925595455265, 19.65208156762590809146416927615, 20.870137971436263426551712310849, 21.72124446257795077041004743214, 22.48295938369164298471721625072, 23.40532064737837069744292588078, 23.63210237153543515573021146059

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