Properties

Label 2-475-475.6-c1-0-29
Degree $2$
Conductor $475$
Sign $-0.281 + 0.959i$
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  + (−2.29 − 1.43i)2-s + (2.30 − 0.660i)3-s + (2.34 + 4.81i)4-s + (2.22 − 0.234i)5-s + (−6.25 − 1.79i)6-s + (1.23 − 2.13i)7-s + (0.950 − 9.04i)8-s + (2.33 − 1.45i)9-s + (−5.45 − 2.65i)10-s + (−2.92 + 3.25i)11-s + (8.58 + 9.53i)12-s + (−0.173 − 4.96i)13-s + (−5.90 + 3.13i)14-s + (4.97 − 2.00i)15-s + (−8.59 + 10.9i)16-s + (−4.87 + 0.340i)17-s + ⋯
L(s)  = 1  + (−1.62 − 1.01i)2-s + (1.33 − 0.381i)3-s + (1.17 + 2.40i)4-s + (0.994 − 0.104i)5-s + (−2.55 − 0.731i)6-s + (0.465 − 0.807i)7-s + (0.336 − 3.19i)8-s + (0.777 − 0.485i)9-s + (−1.72 − 0.840i)10-s + (−0.883 + 0.980i)11-s + (2.47 + 2.75i)12-s + (−0.0481 − 1.37i)13-s + (−1.57 + 0.838i)14-s + (1.28 − 0.518i)15-s + (−2.14 + 2.74i)16-s + (−1.18 + 0.0826i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.281 + 0.959i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.281 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711626 - 0.950499i\)
\(L(\frac12)\) \(\approx\) \(0.711626 - 0.950499i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.22 + 0.234i)T \)
19 \( 1 + (-0.388 + 4.34i)T \)
good2 \( 1 + (2.29 + 1.43i)T + (0.876 + 1.79i)T^{2} \)
3 \( 1 + (-2.30 + 0.660i)T + (2.54 - 1.58i)T^{2} \)
7 \( 1 + (-1.23 + 2.13i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.92 - 3.25i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (0.173 + 4.96i)T + (-12.9 + 0.906i)T^{2} \)
17 \( 1 + (4.87 - 0.340i)T + (16.8 - 2.36i)T^{2} \)
23 \( 1 + (-3.20 + 0.450i)T + (22.1 - 6.33i)T^{2} \)
29 \( 1 + (-5.33 - 0.373i)T + (28.7 + 4.03i)T^{2} \)
31 \( 1 + (-7.59 + 3.38i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-0.586 + 1.80i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (4.18 - 5.36i)T + (-9.91 - 39.7i)T^{2} \)
43 \( 1 + (-0.722 - 4.09i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-0.529 - 0.0369i)T + (46.5 + 6.54i)T^{2} \)
53 \( 1 + (-2.62 - 5.37i)T + (-32.6 + 41.7i)T^{2} \)
59 \( 1 + (1.15 + 2.86i)T + (-42.4 + 40.9i)T^{2} \)
61 \( 1 + (4.50 - 0.632i)T + (58.6 - 16.8i)T^{2} \)
67 \( 1 + (-1.79 - 7.21i)T + (-59.1 + 31.4i)T^{2} \)
71 \( 1 + (1.62 + 1.56i)T + (2.47 + 70.9i)T^{2} \)
73 \( 1 + (0.163 - 4.66i)T + (-72.8 - 5.09i)T^{2} \)
79 \( 1 + (9.15 - 2.62i)T + (66.9 - 41.8i)T^{2} \)
83 \( 1 + (-2.33 + 1.04i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (-5.74 - 7.35i)T + (-21.5 + 86.3i)T^{2} \)
97 \( 1 + (-0.160 + 0.644i)T + (-85.6 - 45.5i)T^{2} \)
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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

−10.41259423445490498644024119668, −9.861246661832578930912827668428, −8.991809007609126768710769625565, −8.259725522088187890548903026668, −7.60661075814283674105886075052, −6.79818854878747222990626997453, −4.60392933754096942268887469403, −2.83200044864395563139598473082, −2.45305620047129593736094351740, −1.13528960683617408391239684618, 1.81830917819265500838939918510, 2.67238021813105759547696712621, 4.96247565514301399100668812833, 6.03265654940541161034510829796, 6.87746641733940484215513023807, 8.124593060091105336884383604558, 8.736473323891309775380411144040, 9.033977837027460151004506807531, 9.986811856939082498003604830232, 10.65180180744226386152809658875

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