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Properties

Label 2-475-95.9-c1-0-16
Degree $2$
Conductor $475$
Sign $0.775 + 0.631i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.708 + 0.124i)2-s + (0.793 + 0.945i)3-s + (−1.39 + 0.507i)4-s + (−0.680 − 0.570i)6-s + (−1.11 − 0.645i)7-s + (2.16 − 1.25i)8-s + (0.256 − 1.45i)9-s + (−2.88 − 4.99i)11-s + (−1.58 − 0.914i)12-s + (1.51 − 1.80i)13-s + (0.873 + 0.317i)14-s + (0.890 − 0.747i)16-s + (6.74 − 1.18i)17-s + 1.06i·18-s + (−2.40 + 3.63i)19-s + ⋯
L(s)  = 1  + (−0.501 + 0.0883i)2-s + (0.458 + 0.545i)3-s + (−0.696 + 0.253i)4-s + (−0.277 − 0.233i)6-s + (−0.422 − 0.244i)7-s + (0.767 − 0.442i)8-s + (0.0854 − 0.484i)9-s + (−0.869 − 1.50i)11-s + (−0.457 − 0.264i)12-s + (0.419 − 0.499i)13-s + (0.233 + 0.0849i)14-s + (0.222 − 0.186i)16-s + (1.63 − 0.288i)17-s + 0.250i·18-s + (−0.551 + 0.833i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.775 + 0.631i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.775 + 0.631i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.864021 - 0.307322i\)
\(L(\frac12)\) \(\approx\) \(0.864021 - 0.307322i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (2.40 - 3.63i)T \)
good2 \( 1 + (0.708 - 0.124i)T + (1.87 - 0.684i)T^{2} \)
3 \( 1 + (-0.793 - 0.945i)T + (-0.520 + 2.95i)T^{2} \)
7 \( 1 + (1.11 + 0.645i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.88 + 4.99i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.51 + 1.80i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-6.74 + 1.18i)T + (15.9 - 5.81i)T^{2} \)
23 \( 1 + (-1.93 - 5.32i)T + (-17.6 + 14.7i)T^{2} \)
29 \( 1 + (-1.04 + 5.92i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-1.19 + 2.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.53iT - 37T^{2} \)
41 \( 1 + (3.51 - 2.94i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.0947 - 0.260i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-0.499 - 0.0880i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + (2.45 + 6.75i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (1.00 + 5.67i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-6.77 + 2.46i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (10.0 + 1.77i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.05 - 0.382i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (1.53 + 1.83i)T + (-12.6 + 71.8i)T^{2} \)
79 \( 1 + (-10.7 + 9.05i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-1.05 - 0.608i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-6.85 - 5.74i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-10.4 + 1.83i)T + (91.1 - 33.1i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.47233095406667408904041115246, −9.943260238176581494417668246599, −9.167739038435613487527548737232, −8.191936039567824247068104275672, −7.77680217161137559424260942281, −6.18414052782433199802127882419, −5.20592654185691244084746958930, −3.67351393728879829973147882142, −3.30241386563325418058722858486, −0.70302019736594817577713518912, 1.51073324451494310126685884136, 2.77846969346495519781250373936, 4.48724343199934359863840464195, 5.27815898743355690803374693866, 6.77087652715222720887405034570, 7.64831950801403975590179124253, 8.442159754843636508813185967439, 9.239458562279606318840120477232, 10.23990042465769581085600734404, 10.65286910763411130070929375330

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