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Properties

Label 1-475-475.6-r0-0-0
Degree $1$
Conductor $475$
Sign $0.918 - 0.395i$
Analytic cond. $2.20589$
Root an. cond. $2.20589$
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 + (0.848 − 0.529i)3-s + (−0.615 + 0.788i)4-s + (0.848 + 0.529i)6-s + (−0.5 − 0.866i)7-s + (−0.978 − 0.207i)8-s + (0.438 − 0.898i)9-s + (−0.104 − 0.994i)11-s + (−0.104 + 0.994i)12-s + (−0.997 + 0.0697i)13-s + (0.559 − 0.829i)14-s + (−0.241 − 0.970i)16-s + (0.990 − 0.139i)17-s + 18-s + (−0.882 − 0.469i)21-s + (0.848 − 0.529i)22-s + ⋯
L(s)  = 1  + (0.438 + 0.898i)2-s + (0.848 − 0.529i)3-s + (−0.615 + 0.788i)4-s + (0.848 + 0.529i)6-s + (−0.5 − 0.866i)7-s + (−0.978 − 0.207i)8-s + (0.438 − 0.898i)9-s + (−0.104 − 0.994i)11-s + (−0.104 + 0.994i)12-s + (−0.997 + 0.0697i)13-s + (0.559 − 0.829i)14-s + (−0.241 − 0.970i)16-s + (0.990 − 0.139i)17-s + 18-s + (−0.882 − 0.469i)21-s + (0.848 − 0.529i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.918 - 0.395i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.918 - 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.751251265 - 0.3605665528i\)
\(L(\frac12)\) \(\approx\) \(1.751251265 - 0.3605665528i\)
\(L(1)\) \(\approx\) \(1.444930927 + 0.1012406667i\)
\(L(1)\) \(\approx\) \(1.444930927 + 0.1012406667i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.438 + 0.898i)T \)
3 \( 1 + (0.848 - 0.529i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (-0.997 + 0.0697i)T \)
17 \( 1 + (0.990 - 0.139i)T \)
23 \( 1 + (0.961 - 0.275i)T \)
29 \( 1 + (0.990 + 0.139i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.241 - 0.970i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.990 + 0.139i)T \)
53 \( 1 + (-0.615 + 0.788i)T \)
59 \( 1 + (-0.719 + 0.694i)T \)
61 \( 1 + (0.961 - 0.275i)T \)
67 \( 1 + (-0.882 + 0.469i)T \)
71 \( 1 + (0.0348 + 0.999i)T \)
73 \( 1 + (-0.997 - 0.0697i)T \)
79 \( 1 + (0.848 - 0.529i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (-0.241 + 0.970i)T \)
97 \( 1 + (-0.882 - 0.469i)T \)
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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

−23.66301907031792231607063267951, −22.72844606079172214537472040175, −21.99417394260228748919731431226, −21.27731682586316655197226473512, −20.60223681892815658267200111146, −19.58473619927702540957301333660, −19.206319448959852541239061749829, −18.23938375203326330086562542709, −17.0706131084374754543900249620, −15.697983229813347070349886172308, −15.05416750895803063293192819315, −14.41381003793054500038395822352, −13.36982744593228423569979215270, −12.47025146061473980139787877624, −11.88041266994726725942809374650, −10.38775594392552881833943520445, −9.864442740198431429445151037711, −9.12761296641127426410247755698, −8.10920160341773178867593020389, −6.75148345774866400082117134394, −5.23759377142385989824667719944, −4.70118637403081848451881814669, −3.332145235707007471501662878129, −2.7223331803919093023614312043, −1.68795879266668179095907253390, 0.79309217663686385990260871776, 2.77847646383307349522009289620, 3.47121769999493800001097233388, 4.576305986058094244067367640935, 5.85979991489065935208232458962, 6.8957047301605621000304045121, 7.49431284536741061867233260564, 8.39722770004281599834931642830, 9.317000395424046740846604394182, 10.31611585301579571561221022134, 11.94711374608535693608255768238, 12.730854275409062284984327822128, 13.64430059718925246191050232288, 14.10835707177461381567270175393, 14.96538709869884341792152205325, 15.968503529676635365592453581208, 16.831327938782778033657972711037, 17.546525158033792073806871195622, 18.84798588763170927801258810351, 19.255723468008384707411636655396, 20.48059075881646758271323666267, 21.23876974630081568857945713823, 22.20852392924899340424504308897, 23.25891913863879812818116181398, 23.788306272830747642897156756378

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