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Properties

Label 1-475-475.204-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.986 - 0.161i$
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  + (−0.997 + 0.0697i)2-s + (0.0348 − 0.999i)3-s + (0.990 − 0.139i)4-s + (0.0348 + 0.999i)6-s + (0.5 − 0.866i)7-s + (−0.978 + 0.207i)8-s + (−0.997 − 0.0697i)9-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)12-s + (0.559 − 0.829i)13-s + (−0.438 + 0.898i)14-s + (0.961 − 0.275i)16-s + (0.374 + 0.927i)17-s + 18-s + (−0.848 − 0.529i)21-s + (0.0348 − 0.999i)22-s + ⋯
L(s)  = 1  + (−0.997 + 0.0697i)2-s + (0.0348 − 0.999i)3-s + (0.990 − 0.139i)4-s + (0.0348 + 0.999i)6-s + (0.5 − 0.866i)7-s + (−0.978 + 0.207i)8-s + (−0.997 − 0.0697i)9-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)12-s + (0.559 − 0.829i)13-s + (−0.438 + 0.898i)14-s + (0.961 − 0.275i)16-s + (0.374 + 0.927i)17-s + 18-s + (−0.848 − 0.529i)21-s + (0.0348 − 0.999i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.986 - 0.161i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.986 - 0.161i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07077576292 - 0.8702719020i\)
\(L(\frac12)\) \(\approx\) \(0.07077576292 - 0.8702719020i\)
\(L(1)\) \(\approx\) \(0.6206706973 - 0.3379569249i\)
\(L(1)\) \(\approx\) \(0.6206706973 - 0.3379569249i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.997 + 0.0697i)T \)
3 \( 1 + (0.0348 - 0.999i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.559 - 0.829i)T \)
17 \( 1 + (0.374 + 0.927i)T \)
23 \( 1 + (0.719 - 0.694i)T \)
29 \( 1 + (0.374 - 0.927i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.961 + 0.275i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.374 - 0.927i)T \)
53 \( 1 + (0.990 - 0.139i)T \)
59 \( 1 + (0.241 - 0.970i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (0.848 - 0.529i)T \)
71 \( 1 + (0.882 - 0.469i)T \)
73 \( 1 + (-0.559 - 0.829i)T \)
79 \( 1 + (-0.0348 + 0.999i)T \)
83 \( 1 + (-0.669 - 0.743i)T \)
89 \( 1 + (-0.961 - 0.275i)T \)
97 \( 1 + (0.848 + 0.529i)T \)
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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

−24.22200110656280236103442775331, −23.20668814531600664751607242157, −21.820345632649891608662668112774, −21.37057132691571565371388435778, −20.73947398023698234786811505486, −19.69824369177669758499493245228, −18.809288876932537746432939130895, −18.1134152077599300421817406861, −17.0753173490238700812207890117, −16.17734920639431016002803076458, −15.7902205259894915579712852260, −14.72782830008409629632763494038, −13.89309434372260954455681523891, −12.23462777388450110052080044143, −11.32666650733359839251545089642, −10.92278018287530125252913174171, −9.7100621644972948710206953954, −8.86538628532198341722680019248, −8.50182607728901829185565085732, −7.14009848821565250786643074666, −5.87800801585904732177996476225, −5.09921327682188427333604546484, −3.52991528903565919971679553530, −2.6975436266953993725323375399, −1.29683089664762008626098193568, 0.33521381410448910978500338070, 1.343138275053302141219749878912, 2.241130408025189868478847034841, 3.599451274231450402270589281012, 5.28104006339740703435296448931, 6.43747222708986392124384521681, 7.22704145045091869963090924899, 7.99410298152935833410868309228, 8.65473741923552035703038374318, 10.07200349634479485340760123693, 10.719084251833392594295248835248, 11.734021696823608913088369884439, 12.631532520320871625398375700351, 13.53514894639032532907777803382, 14.73097727506992619864372773240, 15.371472731978277246680026770925, 16.958786148596809371581945887513, 17.15670877953168609551636313160, 18.13448790411209112055863430580, 18.73514242259809574881718407675, 19.79547663032479270781600056150, 20.34188849155164573253308716544, 21.04320651368337068276968275432, 22.69492083242154816206775949475, 23.460597161484152090821777077331

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