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Properties

Label 2-475-95.44-c1-0-5
Degree $2$
Conductor $475$
Sign $0.231 + 0.972i$
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  + (−1.62 − 1.93i)2-s + (0.223 − 0.613i)3-s + (−0.766 + 4.34i)4-s + (−1.55 + 0.565i)6-s + (−1.32 − 0.766i)7-s + (5.28 − 3.05i)8-s + (1.97 + 1.65i)9-s + (0.592 + 1.02i)11-s + (2.49 + 1.43i)12-s + (0.929 + 2.55i)13-s + (0.673 + 3.82i)14-s + (−6.23 − 2.27i)16-s + (2.49 + 2.97i)17-s − 6.51i·18-s + (−0.819 − 4.28i)19-s + ⋯
L(s)  = 1  + (−1.15 − 1.37i)2-s + (0.128 − 0.354i)3-s + (−0.383 + 2.17i)4-s + (−0.634 + 0.230i)6-s + (−0.501 − 0.289i)7-s + (1.86 − 1.07i)8-s + (0.657 + 0.551i)9-s + (0.178 + 0.309i)11-s + (0.719 + 0.415i)12-s + (0.257 + 0.708i)13-s + (0.180 + 1.02i)14-s + (−1.55 − 0.567i)16-s + (0.604 + 0.720i)17-s − 1.53i·18-s + (−0.187 − 0.982i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.231 + 0.972i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.231 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.637396 - 0.503407i\)
\(L(\frac12)\) \(\approx\) \(0.637396 - 0.503407i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.819 + 4.28i)T \)
good2 \( 1 + (1.62 + 1.93i)T + (-0.347 + 1.96i)T^{2} \)
3 \( 1 + (-0.223 + 0.613i)T + (-2.29 - 1.92i)T^{2} \)
7 \( 1 + (1.32 + 0.766i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.592 - 1.02i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.929 - 2.55i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.49 - 2.97i)T + (-2.95 + 16.7i)T^{2} \)
23 \( 1 + (-4.98 - 0.879i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.56 - 2.99i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.91 + 3.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.10iT - 37T^{2} \)
41 \( 1 + (-9.38 - 3.41i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (8.57 - 1.51i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (0.368 - 0.439i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (2.89 + 0.511i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-3.01 + 2.52i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (0.784 - 4.44i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.49 + 2.97i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.20 + 6.83i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (2.09 - 5.75i)T + (-55.9 - 46.9i)T^{2} \)
79 \( 1 + (-9.21 - 3.35i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-10.6 - 6.15i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.27 - 0.829i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-4.73 - 5.64i)T + (-16.8 + 95.5i)T^{2} \)
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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

−10.76699062152824757254617525540, −9.946712271810197292287802581982, −9.272713362379124640745794348496, −8.367504094442971915706301001672, −7.45655133555851280172938927647, −6.59898622567676748277208469632, −4.63325330531960520907651088366, −3.49726414552924320090235036788, −2.28605661249247930497770021977, −1.12496667491855406498180101074, 0.966744213902868536118628487915, 3.29209889125349117477678312337, 4.88093485388647650742671612520, 5.96247401783312582968187645711, 6.67108621243702982384355810565, 7.63145586447968080608783965288, 8.500614916799073909872626305863, 9.290018596015804239846949865762, 9.964071161861036822401750729559, 10.61247225532722113966769633265

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