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Properties

Label 2-475-19.16-c1-0-16
Degree $2$
Conductor $475$
Sign $0.940 + 0.338i$
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  + (0.484 + 0.406i)2-s + (1.80 − 0.656i)3-s + (−0.277 − 1.57i)4-s + (1.14 + 0.415i)6-s + (2.04 + 3.54i)7-s + (1.13 − 1.97i)8-s + (0.524 − 0.440i)9-s + (2.17 − 3.76i)11-s + (−1.53 − 2.65i)12-s + (1.45 + 0.530i)13-s + (−0.449 + 2.54i)14-s + (−1.65 + 0.601i)16-s + (−4.87 − 4.08i)17-s + 0.433·18-s + (0.708 + 4.30i)19-s + ⋯
L(s)  = 1  + (0.342 + 0.287i)2-s + (1.04 − 0.379i)3-s + (−0.138 − 0.787i)4-s + (0.465 + 0.169i)6-s + (0.772 + 1.33i)7-s + (0.402 − 0.697i)8-s + (0.174 − 0.146i)9-s + (0.655 − 1.13i)11-s + (−0.443 − 0.767i)12-s + (0.404 + 0.147i)13-s + (−0.120 + 0.681i)14-s + (−0.412 + 0.150i)16-s + (−1.18 − 0.991i)17-s + 0.102·18-s + (0.162 + 0.986i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.940 + 0.338i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.940 + 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37656 - 0.414431i\)
\(L(\frac12)\) \(\approx\) \(2.37656 - 0.414431i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.708 - 4.30i)T \)
good2 \( 1 + (-0.484 - 0.406i)T + (0.347 + 1.96i)T^{2} \)
3 \( 1 + (-1.80 + 0.656i)T + (2.29 - 1.92i)T^{2} \)
7 \( 1 + (-2.04 - 3.54i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.17 + 3.76i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.45 - 0.530i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (4.87 + 4.08i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.583 - 3.31i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.99 + 3.35i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (3.28 + 5.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.180T + 37T^{2} \)
41 \( 1 + (0.0242 - 0.00881i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (0.793 - 4.50i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.09 + 0.919i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-0.278 - 1.57i)T + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (-7.31 - 6.13i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.05 + 5.99i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (7.87 - 6.60i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.88 - 10.7i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (12.7 - 4.65i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (16.2 - 5.90i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (2.57 + 4.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.477 - 0.173i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (2.51 + 2.10i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.28262070350063881293547127655, −9.788831884752538078126635424110, −8.855168639870658335634006105731, −8.540180972215327893787018869241, −7.34245459288233810659254743948, −6.08255397874529941506561922077, −5.48529475372692009388564527133, −4.18256091243810215319373586705, −2.74221806925562319355461861248, −1.58587612879602016515598094043, 1.89833258907225783928805425774, 3.25101320384784016642642114383, 4.22483694720642241097724204583, 4.64742324618384355667493407486, 6.77336838303484470077958108586, 7.49595715691780607384762410131, 8.558676348295794472379210251005, 8.953393961509111920492088331578, 10.30842492506095624757234389819, 10.98646666837077385474210241460

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