Properties

Label 1-475-475.461-r0-0-0
Degree $1$
Conductor $475$
Sign $-0.909 + 0.415i$
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  + (−0.374 − 0.927i)2-s + (0.559 + 0.829i)3-s + (−0.719 + 0.694i)4-s + (0.559 − 0.829i)6-s + (−0.5 + 0.866i)7-s + (0.913 + 0.406i)8-s + (−0.374 + 0.927i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.615 − 0.788i)13-s + (0.990 + 0.139i)14-s + (0.0348 − 0.999i)16-s + (−0.241 + 0.970i)17-s + 18-s + (−0.997 + 0.0697i)21-s + (0.559 + 0.829i)22-s + ⋯
L(s)  = 1  + (−0.374 − 0.927i)2-s + (0.559 + 0.829i)3-s + (−0.719 + 0.694i)4-s + (0.559 − 0.829i)6-s + (−0.5 + 0.866i)7-s + (0.913 + 0.406i)8-s + (−0.374 + 0.927i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.615 − 0.788i)13-s + (0.990 + 0.139i)14-s + (0.0348 − 0.999i)16-s + (−0.241 + 0.970i)17-s + 18-s + (−0.997 + 0.0697i)21-s + (0.559 + 0.829i)22-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.909 + 0.415i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ -0.909 + 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06345254724 + 0.2914227070i\)
\(L(\frac12)\) \(\approx\) \(0.06345254724 + 0.2914227070i\)
\(L(1)\) \(\approx\) \(0.6615560313 + 0.05761440541i\)
\(L(1)\) \(\approx\) \(0.6615560313 + 0.05761440541i\)

Euler product

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

−23.7232105763870858523597691415, −23.062618258330306018188973879357, −21.965742725507991625103843688684, −20.62652776706670387510402851495, −19.74172506603202864192680335195, −19.14373195147016244548580282071, −18.18095345057032525507542950696, −17.63408610165807144672613522124, −16.44305785062154624138896247933, −15.91417040465708156388245609715, −14.643949054122308084680104766430, −13.956930206951725560684340273257, −13.34120457567807504672221992780, −12.40959073394705742785034802663, −10.99327051280768688252997419956, −9.84556125662490333202174585486, −9.13400687806509714123359300264, −8.02975278396864723061258109205, −7.27562275189772734173165565499, −6.72727086829883936410266402598, −5.54871044258370760326732223307, −4.34080886893742835107075607680, −3.03432150215256928601969597053, −1.62007539363756959932167748673, −0.16413810171737941957571649761, 2.17636410809285865504134576972, 2.72850923844560613261774619422, 3.80846796198180229189181493010, 4.83963570472601673048652939745, 5.86249539141707670024990076281, 7.79653732284698660381276896363, 8.32135882338029427068548398631, 9.39895237615598609141162454811, 10.06095479702200021492261853504, 10.73367029332210638242301733579, 11.91119079740381220944213257931, 12.83108706528146026616983078226, 13.520081119848252637526266226119, 14.85284202159952361013828429586, 15.47890546801853458019369248554, 16.488762295586868276433640345837, 17.452946626337292703766316895412, 18.46276623874246037212732384904, 19.22226069479574204827120673586, 20.01837098815568167621679395388, 20.73627418508509199531251709931, 21.55658978387448099187853386063, 22.20894497097743822819744641916, 22.84078011499695356526708687444, 24.29818038684246978929953928617

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