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Properties

Label 1-475-475.189-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.728 - 0.684i$
Analytic cond. $51.0458$
Root an. cond. $51.0458$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s − 7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)12-s + (0.309 + 0.951i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)17-s + 18-s + (0.809 + 0.587i)21-s + (−0.809 − 0.587i)22-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.809 + 0.587i)6-s − 7-s + (−0.809 + 0.587i)8-s + (0.309 + 0.951i)9-s + (0.309 − 0.951i)11-s + (0.309 + 0.951i)12-s + (0.309 + 0.951i)13-s + (−0.309 + 0.951i)14-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)17-s + 18-s + (0.809 + 0.587i)21-s + (−0.809 − 0.587i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.728 - 0.684i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4404156718 - 1.112363010i\)
\(L(\frac12)\) \(\approx\) \(0.4404156718 - 1.112363010i\)
\(L(1)\) \(\approx\) \(0.6280138576 - 0.5386264470i\)
\(L(1)\) \(\approx\) \(0.6280138576 - 0.5386264470i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 - T \)
11 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.309 - 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (0.809 + 0.587i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−23.602196118123483838886171478731, −23.02147737185795399262507150536, −22.55303634811412126399664107507, −21.718654487560398719786011514657, −20.81559676115219189561640978964, −19.68901189013388406407579589239, −18.41490030486968824447129518999, −17.69930989318825729044080538665, −16.8578408239318803708062104693, −16.21991194494868429148546452744, −15.36525454072608981372416326805, −14.80684290756239494763834329647, −13.52373715000778341141338669497, −12.47019814961970073881499878164, −12.13489912509574035340936380742, −10.40354672602153197441934005859, −9.89347506029496118415109467327, −8.8360970203818387104079648236, −7.6523426512192929514412397880, −6.508721712199221444529310737287, −6.0252409718432923097581457098, −4.93269772912863837651450649003, −4.03050793736489827140009976956, −3.06692479430383769676801986912, −0.72990175158668665863825443297, 0.51595831616711490694637002626, 1.487894479405613858520286888927, 2.85313668082132392140306111133, 3.83297063527373286448373979552, 5.089392114694349786005690240623, 6.0324211557082602144824722411, 6.75594463264563664305369169421, 8.22298307835394737685885050824, 9.39870348312883047202586982839, 10.18236107942522418013260385560, 11.284145235260157199895589862185, 11.83559873246800907201045279266, 12.64644512581735294789374587046, 13.666370883788172640912676463869, 14.00379124975823791416854826297, 15.67786424792904547649024217167, 16.52936220621840059036221931611, 17.37855838125275957141435436025, 18.60720202805453193456093395372, 18.94378312281029489846041402176, 19.66445380981576186955527256, 20.83980809646142859071856169223, 21.89674819104397426653321029438, 22.200906488986350342960239963372, 23.39889368371393330886093872511

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