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Properties

Label 1-475-475.431-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.536 - 0.843i$
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  + (−0.438 − 0.898i)2-s + (−0.848 + 0.529i)3-s + (−0.615 + 0.788i)4-s + (0.848 + 0.529i)6-s + (−0.5 − 0.866i)7-s + (0.978 + 0.207i)8-s + (0.438 − 0.898i)9-s + (−0.104 − 0.994i)11-s + (0.104 − 0.994i)12-s + (0.997 − 0.0697i)13-s + (−0.559 + 0.829i)14-s + (−0.241 − 0.970i)16-s + (0.990 − 0.139i)17-s − 18-s + (0.882 + 0.469i)21-s + (−0.848 + 0.529i)22-s + ⋯
L(s)  = 1  + (−0.438 − 0.898i)2-s + (−0.848 + 0.529i)3-s + (−0.615 + 0.788i)4-s + (0.848 + 0.529i)6-s + (−0.5 − 0.866i)7-s + (0.978 + 0.207i)8-s + (0.438 − 0.898i)9-s + (−0.104 − 0.994i)11-s + (0.104 − 0.994i)12-s + (0.997 − 0.0697i)13-s + (−0.559 + 0.829i)14-s + (−0.241 − 0.970i)16-s + (0.990 − 0.139i)17-s − 18-s + (0.882 + 0.469i)21-s + (−0.848 + 0.529i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.536 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.536 - 0.843i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.536 - 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4594636467 - 0.8364077354i\)
\(L(\frac12)\) \(\approx\) \(0.4594636467 - 0.8364077354i\)
\(L(1)\) \(\approx\) \(0.5939592693 - 0.2970224419i\)
\(L(1)\) \(\approx\) \(0.5939592693 - 0.2970224419i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.438 - 0.898i)T \)
3 \( 1 + (-0.848 + 0.529i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.104 - 0.994i)T \)
13 \( 1 + (0.997 - 0.0697i)T \)
17 \( 1 + (0.990 - 0.139i)T \)
23 \( 1 + (0.961 - 0.275i)T \)
29 \( 1 + (-0.990 - 0.139i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.241 + 0.970i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.990 + 0.139i)T \)
53 \( 1 + (0.615 - 0.788i)T \)
59 \( 1 + (0.719 - 0.694i)T \)
61 \( 1 + (0.961 - 0.275i)T \)
67 \( 1 + (0.882 - 0.469i)T \)
71 \( 1 + (-0.0348 - 0.999i)T \)
73 \( 1 + (-0.997 - 0.0697i)T \)
79 \( 1 + (-0.848 + 0.529i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (0.241 - 0.970i)T \)
97 \( 1 + (0.882 + 0.469i)T \)
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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

−23.80329010886051605768865921568, −23.20077447929719423184390794562, −22.579025055057808881054066162791, −21.67165730674354489387130031610, −20.36999090788672308764769346397, −19.047437249486517243986766254932, −18.66156456806331348178588714485, −17.91098696059769969196946264868, −17.00736719236122905556211871006, −16.289270668592391035380593207987, −15.45421476156864089242832440706, −14.65551269365279900001649448519, −13.30939989881658155320293331936, −12.74634883126726827298829967780, −11.64552691492403969207128784791, −10.593444927220215164667632618129, −9.63940287694895573692276291073, −8.72273026483245284800907974428, −7.55457652893914598996701604601, −6.89107058300657515014464439317, −5.759069543910615720408905307988, −5.41568979463715967136246604862, −3.99317354625360070032441589997, −2.10604819138487740035755937256, −0.92891916504029128984970083906, 0.47967686260256330687009778984, 1.222386385950155911253481889870, 3.23137402456926993604535256978, 3.716738376698378263728484612988, 4.925040656411413500951883460841, 6.04750604422963527230417903534, 7.201744496400168467328490450696, 8.405692086242841863875399071578, 9.43631406114998074052647463343, 10.248966409701613379606581227639, 11.00756582928090690161676355607, 11.542338340461018881475362138948, 12.8026424314019141048851301598, 13.37345614449666547886791102914, 14.56278017307449243302019758762, 16.09656176018870375961744619596, 16.54561459294239752226361625087, 17.24294418223344592245767318987, 18.377918021269139576354782840223, 18.89230897339597679184741097151, 20.069693848008437144005822958529, 20.843361901478542926645180077554, 21.488314557729545282414135474860, 22.37513538936314970864712285630, 23.1845425303580060160067378715

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