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Properties

Label 2-475-19.11-c1-0-9
Degree $2$
Conductor $475$
Sign $0.449 - 0.893i$
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  + (0.235 − 0.408i)2-s + (−0.520 + 0.900i)3-s + (0.888 + 1.53i)4-s + (0.245 + 0.425i)6-s + 1.17·7-s + 1.78·8-s + (0.958 + 1.66i)9-s + 0.713·11-s − 1.84·12-s + (−2.05 − 3.55i)13-s + (0.277 − 0.480i)14-s + (−1.35 + 2.34i)16-s + (−1.27 + 2.21i)17-s + 0.905·18-s + (1.57 + 4.06i)19-s + ⋯
L(s)  = 1  + (0.166 − 0.288i)2-s + (−0.300 + 0.520i)3-s + (0.444 + 0.769i)4-s + (0.100 + 0.173i)6-s + 0.444·7-s + 0.630·8-s + (0.319 + 0.553i)9-s + 0.215·11-s − 0.533·12-s + (−0.569 − 0.985i)13-s + (0.0741 − 0.128i)14-s + (−0.339 + 0.587i)16-s + (−0.309 + 0.536i)17-s + 0.213·18-s + (0.360 + 0.932i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.449 - 0.893i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.449 - 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39465 + 0.859864i\)
\(L(\frac12)\) \(\approx\) \(1.39465 + 0.859864i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 - 4.06i)T \)
good2 \( 1 + (-0.235 + 0.408i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.520 - 0.900i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.17T + 7T^{2} \)
11 \( 1 - 0.713T + 11T^{2} \)
13 \( 1 + (2.05 + 3.55i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.27 - 2.21i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.303 - 0.525i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.429 - 0.744i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 - 9.38T + 37T^{2} \)
41 \( 1 + (-2.06 + 3.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.06 - 8.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.25 + 9.10i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.49 + 4.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.12 + 5.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.27 - 3.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.59 - 4.48i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.58 + 11.4i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.18 + 10.7i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.98 + 5.16i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + (7.98 + 13.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.35 - 14.4i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.16875795566006939605421518416, −10.44170370303188359086874140095, −9.653985730202349039244456160401, −8.118072482843840217822060167018, −7.81714387136349661696413633808, −6.54415294402549535155733249442, −5.26874160343378359211282368985, −4.36043418241227193102184100381, −3.27731147246231118947269076682, −1.91081745322542058181648413497, 1.08170690215184257773865194515, 2.40906861509876142706981586935, 4.33213738152750648849238929926, 5.21473618646159054673104341523, 6.46611102740461145616776501505, 6.86117298341195383669745068737, 7.83515285492155969346309132869, 9.291299335324386699127312829796, 9.817098590729338979690357670120, 11.22071492552667083934818737964

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