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Properties

Label 2-475-19.16-c1-0-26
Degree $2$
Conductor $475$
Sign $0.276 + 0.961i$
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  + (0.344 + 0.288i)2-s + (1.83 − 0.669i)3-s + (−0.312 − 1.77i)4-s + (0.826 + 0.300i)6-s + (−1.03 − 1.78i)7-s + (0.853 − 1.47i)8-s + (0.635 − 0.533i)9-s + (1.15 − 1.99i)11-s + (−1.75 − 3.04i)12-s + (−4.13 − 1.50i)13-s + (0.160 − 0.911i)14-s + (−2.65 + 0.967i)16-s + (4.19 + 3.51i)17-s + 0.372·18-s + (3.07 − 3.09i)19-s + ⋯
L(s)  = 1  + (0.243 + 0.204i)2-s + (1.06 − 0.386i)3-s + (−0.156 − 0.885i)4-s + (0.337 + 0.122i)6-s + (−0.389 − 0.674i)7-s + (0.301 − 0.522i)8-s + (0.211 − 0.177i)9-s + (0.347 − 0.602i)11-s + (−0.507 − 0.879i)12-s + (−1.14 − 0.417i)13-s + (0.0429 − 0.243i)14-s + (−0.664 + 0.241i)16-s + (1.01 + 0.853i)17-s + 0.0878·18-s + (0.704 − 0.709i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.276 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.276 + 0.961i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.276 + 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60604 - 1.20916i\)
\(L(\frac12)\) \(\approx\) \(1.60604 - 1.20916i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.07 + 3.09i)T \)
good2 \( 1 + (-0.344 - 0.288i)T + (0.347 + 1.96i)T^{2} \)
3 \( 1 + (-1.83 + 0.669i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.03 + 1.78i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.15 + 1.99i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.13 + 1.50i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-4.19 - 3.51i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-1.01 - 5.72i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-4.21 + 3.53i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.378 + 0.656i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.22T + 37T^{2} \)
41 \( 1 + (-6.14 + 2.23i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.549 - 3.11i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (4.87 - 4.08i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.668 - 3.79i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-1.95 - 1.63i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.72 - 9.78i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (1.30 - 1.09i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.71 + 9.70i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-9.60 + 3.49i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (2.26 - 0.824i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (3.53 + 6.11i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.19 - 0.798i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (5.22 + 4.38i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.63012258717379233647983909763, −9.768026152048991831475322749381, −9.185671857120256838984515118341, −7.913325674875983702587316768259, −7.31844071858166102780316693714, −6.15712421592449004569009722550, −5.17863674090341449889220852930, −3.85229556784048172554589236343, −2.73436402176163803386626398984, −1.10662002095850533053320110055, 2.43969038337742284333546417386, 3.09035735246733644500447273332, 4.19118128234925954059110578020, 5.21207602555537073330629798488, 6.80630932088214092813947478214, 7.75947966803354403982559910448, 8.537897537193238703278046771151, 9.473869653771637858014345374265, 9.856615335998032762118105771420, 11.45404869480289995874921721082

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