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Properties

Label 1-475-475.162-r0-0-0
Degree $1$
Conductor $475$
Sign $0.667 + 0.744i$
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  + (−0.788 + 0.615i)2-s + (−0.898 + 0.438i)3-s + (0.241 − 0.970i)4-s + (0.438 − 0.898i)6-s + (0.866 − 0.5i)7-s + (0.406 + 0.913i)8-s + (0.615 − 0.788i)9-s + (−0.978 − 0.207i)11-s + (0.207 + 0.978i)12-s + (−0.139 + 0.990i)13-s + (−0.374 + 0.927i)14-s + (−0.882 − 0.469i)16-s + (−0.275 + 0.961i)17-s + i·18-s + (−0.559 + 0.829i)21-s + (0.898 − 0.438i)22-s + ⋯
L(s)  = 1  + (−0.788 + 0.615i)2-s + (−0.898 + 0.438i)3-s + (0.241 − 0.970i)4-s + (0.438 − 0.898i)6-s + (0.866 − 0.5i)7-s + (0.406 + 0.913i)8-s + (0.615 − 0.788i)9-s + (−0.978 − 0.207i)11-s + (0.207 + 0.978i)12-s + (−0.139 + 0.990i)13-s + (−0.374 + 0.927i)14-s + (−0.882 − 0.469i)16-s + (−0.275 + 0.961i)17-s + i·18-s + (−0.559 + 0.829i)21-s + (0.898 − 0.438i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.667 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.667 + 0.744i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (162, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.667 + 0.744i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6071826662 + 0.2711443640i\)
\(L(\frac12)\) \(\approx\) \(0.6071826662 + 0.2711443640i\)
\(L(1)\) \(\approx\) \(0.5806203552 + 0.1875508512i\)
\(L(1)\) \(\approx\) \(0.5806203552 + 0.1875508512i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.788 + 0.615i)T \)
3 \( 1 + (-0.898 + 0.438i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (-0.139 + 0.990i)T \)
17 \( 1 + (-0.275 + 0.961i)T \)
23 \( 1 + (0.529 - 0.848i)T \)
29 \( 1 + (0.961 - 0.275i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (0.882 + 0.469i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (0.275 + 0.961i)T \)
53 \( 1 + (-0.970 - 0.241i)T \)
59 \( 1 + (0.0348 + 0.999i)T \)
61 \( 1 + (0.848 + 0.529i)T \)
67 \( 1 + (0.829 - 0.559i)T \)
71 \( 1 + (0.997 + 0.0697i)T \)
73 \( 1 + (-0.139 - 0.990i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (0.994 + 0.104i)T \)
89 \( 1 + (-0.882 + 0.469i)T \)
97 \( 1 + (-0.829 - 0.559i)T \)
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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

−23.63834746667271839547879758555, −22.79973478207085509150807095241, −21.85702847038772156652980512159, −21.13385510619701587580607751075, −20.32198275251420859485461431796, −19.22635263412598404164474505077, −18.41136584534087446469829917776, −17.69997394370370992627783968979, −17.426260196994022292164586289225, −15.99409452700043944193034265521, −15.5536552765650823366492891487, −13.903068376520653983866601254237, −12.827417685166575518203287626884, −12.23447340233498565218994853655, −11.27781226996068769173651619771, −10.70340665040034831194941617666, −9.76769934046437988277555600270, −8.48267796367402547878096440163, −7.711882368636863191813753256565, −6.91145087954502160895997539256, −5.42928928168321296012432944926, −4.763587836390061865718685930367, −3.00437576396356261799530720986, −2.015054123592821719343643202805, −0.810477596325663637090602692980, 0.84038099305394235640015586393, 2.15861550753086498963609734782, 4.220506845177876174550118400663, 4.95871774978875781739392674241, 5.97966721965037310609851649586, 6.8651678681016446832099402262, 7.85498848819887974269265740184, 8.80035042523395925621144529563, 9.905052108050145868475827246811, 10.775318538779350621617879021220, 11.17445684378012325980577761842, 12.40789591481580541521142994546, 13.76148215941078548931986209571, 14.71355732513425931907739213227, 15.54077410870803945250190115430, 16.36959523459247430090549646245, 17.12233716216460901461977797397, 17.70901794501014828107057748309, 18.57351645348011227533670532367, 19.4035658721360615180835793484, 20.782160880340377619103357969268, 21.146124537922135464190356810358, 22.41736967851069417206717217111, 23.41841607750984903902312295264, 23.95268373613860461408492833808

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