Properties

Label 2-475-475.111-c1-0-12
Degree $2$
Conductor $475$
Sign $0.124 - 0.992i$
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  + (0.970 + 0.0678i)2-s + (0.0435 + 1.24i)3-s + (−1.04 − 0.146i)4-s + (−0.925 − 2.03i)5-s + (−0.0423 + 1.21i)6-s + (1.73 + 3.00i)7-s + (−2.90 − 0.617i)8-s + (1.44 − 0.100i)9-s + (−0.759 − 2.03i)10-s + (0.537 + 5.11i)11-s + (0.137 − 1.30i)12-s + (1.48 + 2.19i)13-s + (1.47 + 3.03i)14-s + (2.49 − 1.24i)15-s + (−0.751 − 0.215i)16-s + (−0.412 + 1.02i)17-s + ⋯
L(s)  = 1  + (0.686 + 0.0479i)2-s + (0.0251 + 0.719i)3-s + (−0.521 − 0.0733i)4-s + (−0.413 − 0.910i)5-s + (−0.0172 + 0.494i)6-s + (0.655 + 1.13i)7-s + (−1.02 − 0.218i)8-s + (0.480 − 0.0336i)9-s + (−0.240 − 0.644i)10-s + (0.162 + 1.54i)11-s + (0.0396 − 0.377i)12-s + (0.411 + 0.609i)13-s + (0.395 + 0.810i)14-s + (0.644 − 0.320i)15-s + (−0.187 − 0.0538i)16-s + (−0.100 + 0.247i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.124 - 0.992i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20978 + 1.06788i\)
\(L(\frac12)\) \(\approx\) \(1.20978 + 1.06788i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.925 + 2.03i)T \)
19 \( 1 + (0.00711 - 4.35i)T \)
good2 \( 1 + (-0.970 - 0.0678i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (-0.0435 - 1.24i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (-1.73 - 3.00i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.537 - 5.11i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (-1.48 - 2.19i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (0.412 - 1.02i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (0.845 + 0.816i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (2.91 + 7.21i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-5.38 + 5.98i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-4.56 - 3.31i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (9.05 + 2.59i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (-1.13 + 6.45i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.43 - 6.02i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (1.04 + 0.147i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (1.64 + 6.60i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (-3.50 - 3.38i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (4.42 + 2.76i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (-2.03 - 1.08i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (-3.86 + 5.73i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (-0.147 - 4.21i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (-4.22 + 4.68i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (4.28 - 1.22i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-11.0 + 6.90i)T + (42.5 - 87.1i)T^{2} \)
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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

−11.60816651916325909951232859380, −10.01289010638953243331367367446, −9.462934235140919772400585961713, −8.676352952962351159250694415115, −7.78371659477643803384286321545, −6.19361970576963060928554714087, −5.16967385617183343985409568553, −4.44620560922998155006664906677, −3.92728554994428720516735491199, −1.89384326198452449208521392970, 0.880783259710645293291318939021, 3.01615311290981985187789939189, 3.84390982388628829007038924323, 4.93384288309332504110555584889, 6.21795312242381041481757841956, 7.07057774109095005412441074479, 7.955629548580424449613813712704, 8.740594260947689488191713672829, 10.19293416892143132914800724051, 11.01471324912205386719709036880

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