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Properties

Label 1-475-475.203-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: $\chi_{475} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.844 + 0.535i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.184349411 + 0.3436662131i\)
\(L(\frac12)\) \(\approx\) \(1.184349411 + 0.3436662131i\)
\(L(1)\) \(\approx\) \(1.012491505 + 0.02839447962i\)
\(L(1)\) \(\approx\) \(1.012491505 + 0.02839447962i\)

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

−24.024061060114950778546695163627, −23.02328661558599233965989560610, −22.397341171753903579839585724143, −20.718840552162307237220891397024, −20.1239103000273017596911069527, −19.42881813073792660314103179695, −18.6231852449145153272269346049, −17.86072945534400334424624395994, −16.86094824669457506665206466081, −15.90296916463656070214447511268, −15.302969814671324837962897468139, −14.13682764550964667811508082609, −13.69188379464317053707136953046, −12.64274701957122412699366793667, −11.31855380147331162270829652758, −10.03720843626665689091033098854, −9.458048899067403063526650799286, −8.595014552254775316881697257363, −7.69145769403977948341328794194, −6.71652343684283056128289657246, −6.238952998738908166834795468950, −4.51294426168934581347419602263, −3.45267164726712401777470036629, −2.06267069012697205085422734976, −0.83944781572086449474260576923, 1.44430773139154942823038104646, 2.48637998113899724377219869205, 3.62750345672685021353036335035, 4.042274001744715982990096864792, 5.97046942172352477278903415877, 7.085935661810211753011205239333, 8.32528862801799164916723879685, 8.92292153996311661986423306873, 9.58969771201252608314589210480, 10.54011930481054599395140940210, 11.48039076465837926595734259465, 12.58145710076195247417684166940, 13.35239586120005541023508808751, 14.22304856330026290981915647990, 15.54590062337014640619863153798, 16.102145174903753791594724832447, 17.05880117979367134516536537131, 18.19692664568033510412379675232, 19.17274180776714393823842350549, 19.44934952427149388000236899286, 20.37557196636817158529124344712, 21.23940626910621228170201050024, 21.95998188103826579016950813720, 22.58464403923973426763790703398, 24.1500875683193300942902988074

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