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Properties

Label 2-475-95.24-c1-0-12
Degree $2$
Conductor $475$
Sign $0.917 - 0.398i$
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  + (0.300 + 0.826i)2-s + (0.524 + 0.0923i)3-s + (0.939 − 0.788i)4-s + (0.0812 + 0.460i)6-s + (1.62 − 0.939i)7-s + (2.45 + 1.41i)8-s + (−2.55 − 0.929i)9-s + (−1.70 + 2.95i)11-s + (0.565 − 0.326i)12-s + (5.21 − 0.918i)13-s + (1.26 + 1.06i)14-s + (−0.00727 + 0.0412i)16-s + (0.565 + 1.55i)17-s − 2.38i·18-s + (2.52 − 3.55i)19-s + ⋯
L(s)  = 1  + (0.212 + 0.584i)2-s + (0.302 + 0.0533i)3-s + (0.469 − 0.394i)4-s + (0.0331 + 0.188i)6-s + (0.615 − 0.355i)7-s + (0.868 + 0.501i)8-s + (−0.851 − 0.309i)9-s + (−0.514 + 0.890i)11-s + (0.163 − 0.0942i)12-s + (1.44 − 0.254i)13-s + (0.338 + 0.283i)14-s + (−0.00181 + 0.0103i)16-s + (0.137 + 0.376i)17-s − 0.563i·18-s + (0.578 − 0.815i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.917 - 0.398i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.917 - 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05605 + 0.426984i\)
\(L(\frac12)\) \(\approx\) \(2.05605 + 0.426984i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.52 + 3.55i)T \)
good2 \( 1 + (-0.300 - 0.826i)T + (-1.53 + 1.28i)T^{2} \)
3 \( 1 + (-0.524 - 0.0923i)T + (2.81 + 1.02i)T^{2} \)
7 \( 1 + (-1.62 + 0.939i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.70 - 2.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.21 + 0.918i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-0.565 - 1.55i)T + (-13.0 + 10.9i)T^{2} \)
23 \( 1 + (-1.13 - 1.34i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (3.25 + 1.18i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.971 + 1.68i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.837iT - 37T^{2} \)
41 \( 1 + (0.779 - 4.42i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (3.08 - 3.67i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (0.245 - 0.673i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (3.92 + 4.67i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (10.1 - 3.67i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-3.36 + 2.82i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (4.86 - 13.3i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (10.5 + 8.84i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (7.40 + 1.30i)T + (68.5 + 24.9i)T^{2} \)
79 \( 1 + (-1.20 + 6.85i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.17 + 1.25i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.396 - 2.24i)T + (-83.6 + 30.4i)T^{2} \)
97 \( 1 + (0.623 + 1.71i)T + (-74.3 + 62.3i)T^{2} \)
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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

−11.13437853306276237487485903495, −10.28150351076744868116533189706, −9.172026326162856112985278436729, −8.111480108837242047136793281559, −7.47552286220806533051584321274, −6.35278505404692028677520248447, −5.53203003308210214899308153484, −4.50576124036648692150112371092, −3.04610692707993707718089227414, −1.55176973309267922264124066083, 1.64357799668184896537781833160, 2.92758753760774715756995267708, 3.71261166829008653939951857382, 5.24847484919318409485668147126, 6.17044089653468542777083978137, 7.53212953968404775725051439761, 8.291002199781696137511277992941, 8.952698675526493804098750113563, 10.44316770885470883706803970255, 11.14368323373006886385092168859

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