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Properties

Label 1-475-475.161-r0-0-0
Degree $1$
Conductor $475$
Sign $-0.243 + 0.969i$
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  + (0.0348 + 0.999i)2-s + (−0.719 + 0.694i)3-s + (−0.997 + 0.0697i)4-s + (−0.719 − 0.694i)6-s + (−0.5 − 0.866i)7-s + (−0.104 − 0.994i)8-s + (0.0348 − 0.999i)9-s + (0.669 + 0.743i)11-s + (0.669 − 0.743i)12-s + (−0.882 + 0.469i)13-s + (0.848 − 0.529i)14-s + (0.990 − 0.139i)16-s + (0.559 − 0.829i)17-s + 18-s + (0.961 + 0.275i)21-s + (−0.719 + 0.694i)22-s + ⋯
L(s)  = 1  + (0.0348 + 0.999i)2-s + (−0.719 + 0.694i)3-s + (−0.997 + 0.0697i)4-s + (−0.719 − 0.694i)6-s + (−0.5 − 0.866i)7-s + (−0.104 − 0.994i)8-s + (0.0348 − 0.999i)9-s + (0.669 + 0.743i)11-s + (0.669 − 0.743i)12-s + (−0.882 + 0.469i)13-s + (0.848 − 0.529i)14-s + (0.990 − 0.139i)16-s + (0.559 − 0.829i)17-s + 18-s + (0.961 + 0.275i)21-s + (−0.719 + 0.694i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.243 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.243 + 0.969i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ -0.243 + 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5058975435 + 0.6487506855i\)
\(L(\frac12)\) \(\approx\) \(0.5058975435 + 0.6487506855i\)
\(L(1)\) \(\approx\) \(0.6137852756 + 0.4440337963i\)
\(L(1)\) \(\approx\) \(0.6137852756 + 0.4440337963i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.0348 + 0.999i)T \)
3 \( 1 + (-0.719 + 0.694i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (-0.882 + 0.469i)T \)
17 \( 1 + (0.559 - 0.829i)T \)
23 \( 1 + (-0.374 - 0.927i)T \)
29 \( 1 + (0.559 + 0.829i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.990 - 0.139i)T \)
43 \( 1 + (0.766 + 0.642i)T \)
47 \( 1 + (0.559 + 0.829i)T \)
53 \( 1 + (-0.997 + 0.0697i)T \)
59 \( 1 + (-0.615 - 0.788i)T \)
61 \( 1 + (-0.374 - 0.927i)T \)
67 \( 1 + (0.961 - 0.275i)T \)
71 \( 1 + (-0.241 - 0.970i)T \)
73 \( 1 + (-0.882 - 0.469i)T \)
79 \( 1 + (-0.719 + 0.694i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (0.990 + 0.139i)T \)
97 \( 1 + (0.961 + 0.275i)T \)
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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

−23.32578953912229804945411258573, −22.58414117754309399460975188613, −21.8478513109836046503535657964, −21.3601862433708346579638013080, −19.827135394516545151075776067911, −19.246649414126173876591571294833, −18.76348227474280564816721765667, −17.60682255930220648700667212585, −17.15172846785789130492454420744, −15.94389913053375531297340692553, −14.690494560143212611143033500867, −13.70358105477862661909359892097, −12.822141227640107638991457081059, −12.104715145312917901431150430000, −11.584774032198469029931413029076, −10.48926671643467542659506750639, −9.62138316265447882036216359677, −8.54836084178060191663465649487, −7.57926840053981133341091194990, −6.01982340422299135022287496155, −5.62368288622934503119051786387, −4.26488960165226116908750486600, −2.98468306221963934707720706446, −2.02742236555038111363905054682, −0.74016538322370992366393546198, 0.87436553059103557205436190667, 3.215463081713213490764349633847, 4.47448738392846512499131065705, 4.76142038018936706997814370430, 6.23534694973333136483955652457, 6.80673126393931781798481939557, 7.74165803275078221247591731177, 9.271432849513500194537024382589, 9.72809923127127795533098478921, 10.621597321936682898459126063302, 12.04108242525580996944139869146, 12.65499013795954684232213920330, 14.12460788904459585417540090106, 14.51084799006168448397795248533, 15.72578913062873401481412456082, 16.33510763498513016329888674212, 17.09883143826489751105495201682, 17.60284344244908193263860807944, 18.7210221524262398842985353419, 19.826275390516879104214533606444, 20.80220525482817610953004425267, 21.939843190328075173236921702913, 22.559005013809015225802687857697, 23.14239783100003006973036141453, 23.9309989369270576416320508836

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