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Properties

Label 1-475-475.222-r0-0-0
Degree $1$
Conductor $475$
Sign $0.844 + 0.535i$
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  + (0.694 + 0.719i)2-s + (−0.927 − 0.374i)3-s + (−0.0348 + 0.999i)4-s + (−0.374 − 0.927i)6-s + (0.866 − 0.5i)7-s + (−0.743 + 0.669i)8-s + (0.719 + 0.694i)9-s + (0.913 − 0.406i)11-s + (0.406 − 0.913i)12-s + (−0.970 − 0.241i)13-s + (0.961 + 0.275i)14-s + (−0.997 − 0.0697i)16-s + (0.469 − 0.882i)17-s + i·18-s + (−0.990 + 0.139i)21-s + (0.927 + 0.374i)22-s + ⋯
L(s)  = 1  + (0.694 + 0.719i)2-s + (−0.927 − 0.374i)3-s + (−0.0348 + 0.999i)4-s + (−0.374 − 0.927i)6-s + (0.866 − 0.5i)7-s + (−0.743 + 0.669i)8-s + (0.719 + 0.694i)9-s + (0.913 − 0.406i)11-s + (0.406 − 0.913i)12-s + (−0.970 − 0.241i)13-s + (0.961 + 0.275i)14-s + (−0.997 − 0.0697i)16-s + (0.469 − 0.882i)17-s + i·18-s + (−0.990 + 0.139i)21-s + (0.927 + 0.374i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.844 + 0.535i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (222, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.532991525 + 0.4448327386i\)
\(L(\frac12)\) \(\approx\) \(1.532991525 + 0.4448327386i\)
\(L(1)\) \(\approx\) \(1.233140616 + 0.3273492067i\)
\(L(1)\) \(\approx\) \(1.233140616 + 0.3273492067i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.694 + 0.719i)T \)
3 \( 1 + (-0.927 - 0.374i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (-0.970 - 0.241i)T \)
17 \( 1 + (0.469 - 0.882i)T \)
23 \( 1 + (0.829 - 0.559i)T \)
29 \( 1 + (-0.882 + 0.469i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.997 + 0.0697i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.469 - 0.882i)T \)
53 \( 1 + (0.999 + 0.0348i)T \)
59 \( 1 + (0.438 + 0.898i)T \)
61 \( 1 + (0.559 + 0.829i)T \)
67 \( 1 + (0.139 - 0.990i)T \)
71 \( 1 + (0.615 + 0.788i)T \)
73 \( 1 + (-0.970 + 0.241i)T \)
79 \( 1 + (-0.374 + 0.927i)T \)
83 \( 1 + (0.207 + 0.978i)T \)
89 \( 1 + (-0.997 + 0.0697i)T \)
97 \( 1 + (-0.139 - 0.990i)T \)
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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

−23.55445605545958372859738455565, −22.724210826485209951245079117608, −21.98521868218887563328330602742, −21.36810335187697153721289334438, −20.69979015099582085694137828655, −19.50233767018873150585696872782, −18.8028036360522728806676612128, −17.59557518127558706380536105458, −17.140079589567821830761811905799, −15.79463327046246302480516657593, −14.8273204453873317318792508460, −14.48430139514393039165005105439, −12.98463570351342653463070016021, −12.16211698070210453804152822800, −11.61097988848437978547942843657, −10.8240098591327846561498878208, −9.814133611993881195621011761894, −9.07332804013026210550215908506, −7.39803444832452619875753495000, −6.24413845319469306765536462612, −5.39502932794355630134468469144, −4.58431512535687819606787500114, −3.77188158007421259822200775188, −2.217994004707364066807865267735, −1.16788896569123285105819119540, 1.02139521294327278351917132682, 2.64465367993065955430227195559, 4.16147731957470195383804402936, 4.91709909870791259443851283469, 5.73214151434956866710848633932, 6.869323410652756812126296627611, 7.414105398049470127835318252505, 8.42928679999241793319564492059, 9.80414274968670849266050513052, 11.176569600357154611242615021766, 11.701724239280250292878742537870, 12.588605001161155898816957810280, 13.54306026514617182345237098189, 14.38941550198668583062369152160, 15.12802504273058673247559066713, 16.52519086011085483587031384317, 16.80612149434993639162303244379, 17.64688980584519255853293596888, 18.43215501584381814735980658627, 19.66644889405774061087872521233, 20.83159262650958143135967982541, 21.593890882538044837606049563794, 22.54949987850839698927823809639, 22.944541586092912840277796473060, 24.01867030500788265705682502787

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