Properties

Label 2-475-475.111-c1-0-18
Degree $2$
Conductor $475$
Sign $0.795 - 0.605i$
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  + (−0.339 − 0.0237i)2-s + (0.0489 + 1.40i)3-s + (−1.86 − 0.262i)4-s + (1.65 − 1.50i)5-s + (0.0166 − 0.477i)6-s + (1.82 + 3.16i)7-s + (1.29 + 0.274i)8-s + (1.02 − 0.0719i)9-s + (−0.597 + 0.471i)10-s + (−0.510 − 4.85i)11-s + (0.276 − 2.62i)12-s + (−1.77 − 2.62i)13-s + (−0.544 − 1.11i)14-s + (2.19 + 2.24i)15-s + (3.18 + 0.914i)16-s + (−2.69 + 6.67i)17-s + ⋯
L(s)  = 1  + (−0.240 − 0.0167i)2-s + (0.0282 + 0.809i)3-s + (−0.932 − 0.131i)4-s + (0.739 − 0.672i)5-s + (0.00680 − 0.194i)6-s + (0.690 + 1.19i)7-s + (0.457 + 0.0971i)8-s + (0.343 − 0.0239i)9-s + (−0.188 + 0.149i)10-s + (−0.153 − 1.46i)11-s + (0.0797 − 0.758i)12-s + (−0.491 − 0.728i)13-s + (−0.145 − 0.298i)14-s + (0.565 + 0.579i)15-s + (0.797 + 0.228i)16-s + (−0.654 + 1.61i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.795 - 0.605i$
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: \((2,\ 475,\ (\ :1/2),\ 0.795 - 0.605i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23596 + 0.416750i\)
\(L(\frac12)\) \(\approx\) \(1.23596 + 0.416750i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.65 + 1.50i)T \)
19 \( 1 + (-4.30 + 0.652i)T \)
good2 \( 1 + (0.339 + 0.0237i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (-0.0489 - 1.40i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (-1.82 - 3.16i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.510 + 4.85i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (1.77 + 2.62i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (2.69 - 6.67i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (-4.81 - 4.65i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (-2.27 - 5.61i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-3.60 + 4.00i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-6.37 - 4.62i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.43 + 0.410i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (0.199 - 1.13i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.50 + 6.19i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (7.12 + 1.00i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (-1.65 - 6.62i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (-1.97 - 1.90i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (0.280 + 0.175i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (14.1 + 7.51i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (0.666 - 0.987i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.397 + 11.3i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (-9.97 + 11.0i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (4.32 - 1.24i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-6.51 + 4.06i)T + (42.5 - 87.1i)T^{2} \)
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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

−10.81524116531858004954430491649, −10.07476020155546126639870834092, −9.179879483266661088400711327589, −8.709302670644527421649094657836, −7.970011636402305936900335356199, −5.96879105349227164687643686147, −5.29399082022177294156376623998, −4.63324542474433565505106547335, −3.20638808600903249317432399719, −1.36032256096327660407457333802, 1.14334946022471971906853681194, 2.49786756485141901364999176768, 4.41413369198147818850080246501, 4.89402174796154298434184186617, 6.78623945224626965064930550649, 7.20612304889651130998792955806, 7.906954054441994199637772562167, 9.457977079000218024289905832855, 9.788983355827692322413106390443, 10.75312822616201047172059123416

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