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Properties

Label 2-475-19.5-c1-0-27
Degree $2$
Conductor $475$
Sign $-0.753 + 0.657i$
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  + (1.58 − 0.575i)2-s + (0.564 − 3.20i)3-s + (0.639 − 0.536i)4-s + (−0.950 − 5.39i)6-s + (−0.274 − 0.474i)7-s + (−0.981 + 1.69i)8-s + (−7.11 − 2.59i)9-s + (−0.165 + 0.286i)11-s + (−1.35 − 2.35i)12-s + (−0.837 − 4.74i)13-s + (−0.707 − 0.593i)14-s + (−0.863 + 4.89i)16-s + (4.96 − 1.80i)17-s − 12.7·18-s + (4.30 − 0.670i)19-s + ⋯
L(s)  = 1  + (1.11 − 0.407i)2-s + (0.326 − 1.84i)3-s + (0.319 − 0.268i)4-s + (−0.388 − 2.20i)6-s + (−0.103 − 0.179i)7-s + (−0.346 + 0.600i)8-s + (−2.37 − 0.863i)9-s + (−0.0499 + 0.0864i)11-s + (−0.391 − 0.678i)12-s + (−0.232 − 1.31i)13-s + (−0.189 − 0.158i)14-s + (−0.215 + 1.22i)16-s + (1.20 − 0.438i)17-s − 3.00·18-s + (0.988 − 0.153i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.753 + 0.657i$
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: \((2,\ 475,\ (\ :1/2),\ -0.753 + 0.657i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.846420 - 2.25902i\)
\(L(\frac12)\) \(\approx\) \(0.846420 - 2.25902i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.30 + 0.670i)T \)
good2 \( 1 + (-1.58 + 0.575i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (-0.564 + 3.20i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (0.274 + 0.474i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.165 - 0.286i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.837 + 4.74i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-4.96 + 1.80i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-0.850 + 0.713i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-3.01 - 1.09i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (-3.01 - 5.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.67T + 37T^{2} \)
41 \( 1 + (1.37 - 7.79i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.25 + 1.05i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (4.32 + 1.57i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (5.15 - 4.32i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (-6.39 + 2.32i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.520 + 0.436i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (7.30 + 2.65i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.832 + 0.698i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (2.42 - 13.7i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (0.243 - 1.38i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (0.427 + 0.740i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.52 - 14.3i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (6.59 - 2.40i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.20353953799171963622584013886, −9.844059669519358566292139442819, −8.443855403552664334456803368516, −7.85427205284560996780984071340, −6.93551268842094952895665920105, −5.86460675574188007046510026403, −5.08748182493056324298827320995, −3.24184193529301680352356482053, −2.73948557633750869008696823519, −1.10246872293603897459210521959, 2.92202646353229426798432589055, 3.83778165118899910129896727707, 4.57087247489546540775813324762, 5.41130857256209026731708329384, 6.21780346848805219770702213303, 7.71947652499771407695307231873, 8.988263311931776272020725022897, 9.627434606919522193631538519291, 10.23510575264210156363907125656, 11.47489510248447410924786772745

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