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Properties

Label 2-475-19.5-c1-0-7
Degree $2$
Conductor $475$
Sign $0.764 - 0.644i$
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  + (−1.89 + 0.691i)2-s + (0.330 − 1.87i)3-s + (1.59 − 1.34i)4-s + (0.667 + 3.78i)6-s + (1.54 + 2.67i)7-s + (−0.0878 + 0.152i)8-s + (−0.575 − 0.209i)9-s + (−0.481 + 0.834i)11-s + (−1.98 − 3.43i)12-s + (0.513 + 2.91i)13-s + (−4.78 − 4.01i)14-s + (−0.663 + 3.76i)16-s + (0.0366 − 0.0133i)17-s + 1.23·18-s + (−4.31 + 0.596i)19-s + ⋯
L(s)  = 1  + (−1.34 + 0.488i)2-s + (0.190 − 1.08i)3-s + (0.799 − 0.670i)4-s + (0.272 + 1.54i)6-s + (0.584 + 1.01i)7-s + (−0.0310 + 0.0538i)8-s + (−0.191 − 0.0697i)9-s + (−0.145 + 0.251i)11-s + (−0.572 − 0.991i)12-s + (0.142 + 0.808i)13-s + (−1.27 − 1.07i)14-s + (−0.165 + 0.940i)16-s + (0.00887 − 0.00323i)17-s + 0.291·18-s + (−0.990 + 0.136i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.764 - 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.730018 + 0.266425i\)
\(L(\frac12)\) \(\approx\) \(0.730018 + 0.266425i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.31 - 0.596i)T \)
good2 \( 1 + (1.89 - 0.691i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (-0.330 + 1.87i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (-1.54 - 2.67i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.481 - 0.834i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.513 - 2.91i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.0366 + 0.0133i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (2.97 - 2.49i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-8.76 - 3.19i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-4.68 - 8.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.11T + 37T^{2} \)
41 \( 1 + (-2.09 + 11.8i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-3.79 - 3.18i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-7.53 - 2.74i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (4.64 - 3.89i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (2.60 - 0.948i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.68 - 6.44i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-8.78 - 3.19i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-9.64 - 8.09i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (0.155 - 0.883i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.54 + 8.75i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (4.80 + 8.31i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.51 + 8.61i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (-7.31 + 2.66i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.89225491109865137703809634447, −10.03448134907838416191023035676, −8.827400745741555647640226309475, −8.527091832790592722891065198328, −7.59059774518550295189080851974, −6.82978825926969099565858611624, −6.00688381133578841068331727265, −4.47368786649123273912131039638, −2.35092632309347597776395399539, −1.37317954870420532438775063445, 0.828379384721487861194979713170, 2.58241905180464856364199891563, 4.04666178046810551711350523232, 4.85165681839296785396919403120, 6.46060398624066958106733025081, 7.937057236757886484016320989955, 8.199230575445727225330812167162, 9.395631486757647195455382638550, 10.05381644904678841015615853106, 10.67318700312973993044651380754

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