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Properties

Label 1-475-475.111-r0-0-0
Degree $1$
Conductor $475$
Sign $-0.560 + 0.828i$
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  + (0.990 + 0.139i)2-s + (−0.997 + 0.0697i)3-s + (0.961 + 0.275i)4-s + (−0.997 − 0.0697i)6-s + (−0.5 + 0.866i)7-s + (0.913 + 0.406i)8-s + (0.990 − 0.139i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.374 + 0.927i)13-s + (−0.615 + 0.788i)14-s + (0.848 + 0.529i)16-s + (−0.719 − 0.694i)17-s + 18-s + (0.438 − 0.898i)21-s + (−0.997 + 0.0697i)22-s + ⋯
L(s)  = 1  + (0.990 + 0.139i)2-s + (−0.997 + 0.0697i)3-s + (0.961 + 0.275i)4-s + (−0.997 − 0.0697i)6-s + (−0.5 + 0.866i)7-s + (0.913 + 0.406i)8-s + (0.990 − 0.139i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.374 + 0.927i)13-s + (−0.615 + 0.788i)14-s + (0.848 + 0.529i)16-s + (−0.719 − 0.694i)17-s + 18-s + (0.438 − 0.898i)21-s + (−0.997 + 0.0697i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.560 + 0.828i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ -0.560 + 0.828i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5872289386 + 1.106004444i\)
\(L(\frac12)\) \(\approx\) \(0.5872289386 + 1.106004444i\)
\(L(1)\) \(\approx\) \(1.083743782 + 0.4757277348i\)
\(L(1)\) \(\approx\) \(1.083743782 + 0.4757277348i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.990 + 0.139i)T \)
3 \( 1 + (-0.997 + 0.0697i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (-0.374 + 0.927i)T \)
17 \( 1 + (-0.719 - 0.694i)T \)
23 \( 1 + (0.0348 + 0.999i)T \)
29 \( 1 + (-0.719 + 0.694i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.848 + 0.529i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.719 + 0.694i)T \)
53 \( 1 + (0.961 + 0.275i)T \)
59 \( 1 + (-0.882 + 0.469i)T \)
61 \( 1 + (0.0348 + 0.999i)T \)
67 \( 1 + (0.438 + 0.898i)T \)
71 \( 1 + (0.559 + 0.829i)T \)
73 \( 1 + (-0.374 - 0.927i)T \)
79 \( 1 + (-0.997 + 0.0697i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (0.848 - 0.529i)T \)
97 \( 1 + (0.438 - 0.898i)T \)
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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

−23.10252430632814686198872780277, −22.99829128953046432464352297037, −21.9694840483993133480700097751, −21.23575873340530894121060442529, −20.25144359050944439519949825071, −19.488906880824238819379515880213, −18.36952328519543144908236253725, −17.34257889282611868796227028316, −16.50714887366362814906307078334, −15.78986277867430611856688545809, −14.955265666861824818276546479723, −13.694056110695864399377790173008, −12.86895283537075255129285878287, −12.52559222897192085388379557928, −11.1301165414991474526505585053, −10.63043791191978410815974322068, −9.92275533042105253557260456419, −7.96821384022877678472524876942, −7.05340412693688710928545430616, −6.2170292651751358038797026270, −5.31968445658655096539427994036, −4.44760918316308838624094448879, −3.41980682152757144201135261609, −2.11203758073882374793944633501, −0.52473755973612859345777161286, 1.852103563488171271116064563299, 2.91580596772945985757688816701, 4.28987115396611531639884032313, 5.13128065864817964629668075474, 5.87502311870162551171715748382, 6.79654862305847374044937748582, 7.616951489066477467969719993177, 9.22522820818433543672119852767, 10.22366795916987360466953074922, 11.4395466393824993808911812118, 11.77211057084563925545422215588, 12.91133194667832780109507909259, 13.35000749754730115969542584667, 14.78471908118875242699740152043, 15.57883851668829969596124470804, 16.17015628788471303242516912041, 16.99009920407620172506931315236, 18.12730322259652654682276054963, 18.89380420595908462700577519663, 20.065428197156791628247660023057, 21.16614323320456450810528781458, 21.742839789115570586120676797331, 22.42850588910634229657160324851, 23.173555755852406176952369427821, 24.00557708612125526483090085923

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