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Properties

Label 2-475-19.6-c1-0-18
Degree $2$
Conductor $475$
Sign $0.996 - 0.0849i$
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  + (1.20 − 1.00i)2-s + (2.14 + 0.781i)3-s + (0.0809 − 0.459i)4-s + (3.37 − 1.22i)6-s + (−2.00 + 3.47i)7-s + (1.20 + 2.08i)8-s + (1.70 + 1.42i)9-s + (−1.38 − 2.39i)11-s + (0.532 − 0.922i)12-s + (2.61 − 0.953i)13-s + (1.09 + 6.20i)14-s + (4.43 + 1.61i)16-s + (2.76 − 2.31i)17-s + 3.48·18-s + (1.79 − 3.97i)19-s + ⋯
L(s)  = 1  + (0.850 − 0.713i)2-s + (1.23 + 0.451i)3-s + (0.0404 − 0.229i)4-s + (1.37 − 0.501i)6-s + (−0.758 + 1.31i)7-s + (0.425 + 0.737i)8-s + (0.567 + 0.475i)9-s + (−0.417 − 0.722i)11-s + (0.153 − 0.266i)12-s + (0.726 − 0.264i)13-s + (0.292 + 1.65i)14-s + (1.10 + 0.403i)16-s + (0.670 − 0.562i)17-s + 0.822·18-s + (0.410 − 0.911i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.996 - 0.0849i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.996 - 0.0849i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.96709 + 0.126253i\)
\(L(\frac12)\) \(\approx\) \(2.96709 + 0.126253i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.79 + 3.97i)T \)
good2 \( 1 + (-1.20 + 1.00i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (-2.14 - 0.781i)T + (2.29 + 1.92i)T^{2} \)
7 \( 1 + (2.00 - 3.47i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.38 + 2.39i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.61 + 0.953i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.76 + 2.31i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (0.237 - 1.34i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (7.28 + 6.11i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.776 - 1.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.51T + 37T^{2} \)
41 \( 1 + (-6.21 - 2.26i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.08 - 6.15i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (6.97 + 5.85i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (-0.684 + 3.88i)T + (-49.8 - 18.1i)T^{2} \)
59 \( 1 + (-2.76 + 2.32i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.30 - 7.37i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-11.1 - 9.36i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.576 + 3.26i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (9.40 + 3.42i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-1.82 - 0.666i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-0.809 + 1.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-11.5 + 4.19i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (-8.87 + 7.45i)T + (16.8 - 95.5i)T^{2} \)
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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

−11.30804948904833000772849316965, −10.06111561853630893237293164026, −9.174862322030862543375817714693, −8.551866575613164880190462287693, −7.66761126402812725204134627949, −5.96970774350671868624432463472, −5.14900615477294869596268494451, −3.66884254015895115775594357837, −3.11271305769052397234882570532, −2.33216724826300767997507996256, 1.57031920451524189256953018699, 3.45829580480344371959996553923, 3.91471972353878179355120168836, 5.36356824215108729426272750082, 6.51717116046396632388948755955, 7.34936418851470307863793850245, 7.83894060239302040559051226895, 9.143414611991140371133954525870, 10.04654861467335769516301676445, 10.75209768355603454110536017028

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