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Properties

Label 2-475-475.111-c1-0-40
Degree $2$
Conductor $475$
Sign $-0.564 - 0.825i$
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  + (−1.84 − 0.129i)2-s + (−0.0736 − 2.10i)3-s + (1.41 + 0.198i)4-s + (−1.96 − 1.05i)5-s + (−0.136 + 3.90i)6-s + (−2.35 − 4.08i)7-s + (1.03 + 0.220i)8-s + (−1.45 + 0.101i)9-s + (3.50 + 2.21i)10-s + (−0.430 − 4.09i)11-s + (0.315 − 2.99i)12-s + (−0.277 − 0.411i)13-s + (3.82 + 7.85i)14-s + (−2.08 + 4.23i)15-s + (−4.62 − 1.32i)16-s + (2.73 − 6.76i)17-s + ⋯
L(s)  = 1  + (−1.30 − 0.0913i)2-s + (−0.0425 − 1.21i)3-s + (0.707 + 0.0993i)4-s + (−0.880 − 0.473i)5-s + (−0.0556 + 1.59i)6-s + (−0.891 − 1.54i)7-s + (0.366 + 0.0778i)8-s + (−0.484 + 0.0338i)9-s + (1.10 + 0.699i)10-s + (−0.129 − 1.23i)11-s + (0.0909 − 0.865i)12-s + (−0.0770 − 0.114i)13-s + (1.02 + 2.09i)14-s + (−0.539 + 1.09i)15-s + (−1.15 − 0.331i)16-s + (0.663 − 1.64i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.564 - 0.825i$
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: \((2,\ 475,\ (\ :1/2),\ -0.564 - 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.177623 + 0.336676i\)
\(L(\frac12)\) \(\approx\) \(0.177623 + 0.336676i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.96 + 1.05i)T \)
19 \( 1 + (2.38 - 3.64i)T \)
good2 \( 1 + (1.84 + 0.129i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (0.0736 + 2.10i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (2.35 + 4.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.430 + 4.09i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (0.277 + 0.411i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (-2.73 + 6.76i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (-0.906 - 0.875i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (-0.273 - 0.676i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-2.13 + 2.36i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (-4.68 - 3.40i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.88 - 2.26i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (1.20 - 6.85i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.41 + 5.96i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (0.631 + 0.0887i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (-0.698 - 2.80i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (-5.89 - 5.69i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (6.26 + 3.91i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (6.00 + 3.19i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (3.52 - 5.22i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (0.316 + 9.06i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (-9.25 + 10.2i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-9.78 + 2.80i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-5.41 + 3.38i)T + (42.5 - 87.1i)T^{2} \)
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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

−10.30166518036347987289345870930, −9.502694507017974841346867476746, −8.398951330723009242343881586404, −7.61974095925376097187139343496, −7.30820911030844035014639723705, −6.22418070225795647324626017642, −4.46549488867996074889919488258, −3.13633373190339545276351349324, −1.05978191061720611344759069285, −0.45657415970588570905659785313, 2.38999531235861835811982359308, 3.80378521754422610648624871749, 4.78866449267776787202521325866, 6.22133266541648094211303094468, 7.26326100313314987801309061737, 8.331968900250046914434155993674, 9.056538198327322192640452505516, 9.741389668153507287809802424500, 10.40137826337595737087828387000, 11.12463452696659579902470906740

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