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Properties

Label 2-475-95.4-c1-0-13
Degree $2$
Conductor $475$
Sign $0.139 - 0.990i$
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  + (−0.408 + 1.12i)2-s + (2.23 − 0.394i)3-s + (0.439 + 0.369i)4-s + (−0.471 + 2.67i)6-s + (1.90 + 1.09i)7-s + (−2.66 + 1.53i)8-s + (2.03 − 0.741i)9-s + (1.41 + 2.44i)11-s + (1.13 + 0.652i)12-s + (−4.01 − 0.708i)13-s + (−2.00 + 1.68i)14-s + (−0.437 − 2.48i)16-s + (0.130 − 0.359i)17-s + 2.58i·18-s + (2.75 − 3.37i)19-s + ⋯
L(s)  = 1  + (−0.288 + 0.793i)2-s + (1.29 − 0.227i)3-s + (0.219 + 0.184i)4-s + (−0.192 + 1.09i)6-s + (0.718 + 0.415i)7-s + (−0.941 + 0.543i)8-s + (0.678 − 0.247i)9-s + (0.426 + 0.738i)11-s + (0.326 + 0.188i)12-s + (−1.11 − 0.196i)13-s + (−0.536 + 0.450i)14-s + (−0.109 − 0.620i)16-s + (0.0317 − 0.0872i)17-s + 0.609i·18-s + (0.632 − 0.774i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.139 - 0.990i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.139 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53473 + 1.33406i\)
\(L(\frac12)\) \(\approx\) \(1.53473 + 1.33406i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.75 + 3.37i)T \)
good2 \( 1 + (0.408 - 1.12i)T + (-1.53 - 1.28i)T^{2} \)
3 \( 1 + (-2.23 + 0.394i)T + (2.81 - 1.02i)T^{2} \)
7 \( 1 + (-1.90 - 1.09i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.41 - 2.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.01 + 0.708i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-0.130 + 0.359i)T + (-13.0 - 10.9i)T^{2} \)
23 \( 1 + (1.91 - 2.27i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-8.05 + 2.93i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.34 + 4.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.7iT - 37T^{2} \)
41 \( 1 + (-0.544 - 3.08i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (1.29 + 1.53i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (-3.72 - 10.2i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + (-4.80 + 5.72i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (12.1 + 4.43i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-4.57 - 3.84i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (0.358 + 0.986i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (5.99 - 5.03i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 + (13.8 - 2.44i)T + (68.5 - 24.9i)T^{2} \)
79 \( 1 + (-2.33 - 13.2i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (4.81 + 2.77i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.13 + 6.46i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (-4.94 + 13.5i)T + (-74.3 - 62.3i)T^{2} \)
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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

−11.41998808652017992945070314696, −9.859581243992467751730435907257, −9.141401367396935722589461319906, −8.337248170254210239309383304025, −7.60139502008502368256424749011, −7.09183359720380353683583398710, −5.74666740537216258646906988158, −4.50561949647505017348475203301, −2.92530690005356696151695496354, −2.14393656809318588433249889376, 1.37916116462490566913520943827, 2.63868993531935655104727331587, 3.47464153157157606236951688118, 4.73894235136718300723824973321, 6.23626141514185197509352973096, 7.41322453815267195981353718755, 8.387297255280077405341255009085, 9.047721506070613087024602211995, 10.07794533320173456478368130341, 10.48761184509171231129365816552

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