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Properties

Label 1-475-475.137-r1-0-0
Degree $1$
Conductor $475$
Sign $0.909 + 0.415i$
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  + (−0.927 − 0.374i)2-s + (−0.829 − 0.559i)3-s + (0.719 + 0.694i)4-s + (0.559 + 0.829i)6-s + (0.866 − 0.5i)7-s + (−0.406 − 0.913i)8-s + (0.374 + 0.927i)9-s + (−0.978 − 0.207i)11-s + (−0.207 − 0.978i)12-s + (0.788 + 0.615i)13-s + (−0.990 + 0.139i)14-s + (0.0348 + 0.999i)16-s + (0.970 − 0.241i)17-s − i·18-s + (−0.997 − 0.0697i)21-s + (0.829 + 0.559i)22-s + ⋯
L(s)  = 1  + (−0.927 − 0.374i)2-s + (−0.829 − 0.559i)3-s + (0.719 + 0.694i)4-s + (0.559 + 0.829i)6-s + (0.866 − 0.5i)7-s + (−0.406 − 0.913i)8-s + (0.374 + 0.927i)9-s + (−0.978 − 0.207i)11-s + (−0.207 − 0.978i)12-s + (0.788 + 0.615i)13-s + (−0.990 + 0.139i)14-s + (0.0348 + 0.999i)16-s + (0.970 − 0.241i)17-s − i·18-s + (−0.997 − 0.0697i)21-s + (0.829 + 0.559i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.909 + 0.415i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.909 + 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8213061084 + 0.1788260262i\)
\(L(\frac12)\) \(\approx\) \(0.8213061084 + 0.1788260262i\)
\(L(1)\) \(\approx\) \(0.6035821808 - 0.1241404537i\)
\(L(1)\) \(\approx\) \(0.6035821808 - 0.1241404537i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.927 - 0.374i)T \)
3 \( 1 + (-0.829 - 0.559i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-0.978 - 0.207i)T \)
13 \( 1 + (0.788 + 0.615i)T \)
17 \( 1 + (0.970 - 0.241i)T \)
23 \( 1 + (0.469 + 0.882i)T \)
29 \( 1 + (0.241 - 0.970i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (0.0348 + 0.999i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.970 - 0.241i)T \)
53 \( 1 + (-0.694 + 0.719i)T \)
59 \( 1 + (-0.848 + 0.529i)T \)
61 \( 1 + (-0.882 + 0.469i)T \)
67 \( 1 + (-0.0697 - 0.997i)T \)
71 \( 1 + (0.438 + 0.898i)T \)
73 \( 1 + (-0.788 + 0.615i)T \)
79 \( 1 + (-0.559 + 0.829i)T \)
83 \( 1 + (0.994 + 0.104i)T \)
89 \( 1 + (-0.0348 + 0.999i)T \)
97 \( 1 + (0.0697 - 0.997i)T \)
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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

−23.58211354042784600554883641196, −22.984834265238647539496445954124, −21.69785850287319880821584349524, −20.825777260504810878450209827258, −20.42242742009274579242511110617, −18.82080600766658103980534553138, −18.25340797396020688279676905090, −17.64757773460070529484700139892, −16.68518777987109822725128362863, −15.969251391433250034182483336, −15.14595717792791925946561566768, −14.53337643613431478188472710792, −12.86869320316676861472296167807, −11.86522699054958472251200991287, −10.90821591221087363263003634728, −10.46610846473353116561165306191, −9.41377439282973141860240882848, −8.37507927251611518123557926541, −7.62412945596212028486168679796, −6.31093669382107148042900276429, −5.5116094168169298958517968326, −4.784310929877268415064163502998, −3.08138561089922314203138304187, −1.61325501662507435056166145841, −0.41986360015269661019101358853, 0.96756688479231246413536664787, 1.67371466355810810279977898321, 3.070002043069980919003046347354, 4.542349842745454218961118087556, 5.70412953325167283780548636417, 6.80550277539932714535672381569, 7.744805922480495094420823463991, 8.26476657011311747920045117901, 9.703737161119176721114169928215, 10.63968883882874489224267916943, 11.29190789753000933036032635402, 11.944759688992068889669015907708, 13.0754373172687948204781185053, 13.84146372283589598844510357498, 15.34679852187554820637727986483, 16.36576169286558896497418876916, 16.90465114772131186402235175863, 17.90021255981142839310525248847, 18.37510466144884436478646042004, 19.13675053559001281205712353441, 20.175444708128328642545976969315, 21.29300567433015303763569625409, 21.4717416637209034947899939588, 23.19342205344823187800632010344, 23.52762153539011234818730266207

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