Properties

Label 2-475-5.4-c1-0-19
Degree $2$
Conductor $475$
Sign $-0.894 + 0.447i$
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  − 2.37i·2-s + 1.27i·3-s − 3.65·4-s + 3.02·6-s − 0.726i·7-s + 3.92i·8-s + 1.37·9-s − 0.273·11-s − 4.65i·12-s − 5.95i·13-s − 1.72·14-s + 2.02·16-s − 5.27i·17-s − 3.27i·18-s − 19-s + ⋯
L(s)  = 1  − 1.68i·2-s + 0.735i·3-s − 1.82·4-s + 1.23·6-s − 0.274i·7-s + 1.38i·8-s + 0.459·9-s − 0.0825·11-s − 1.34i·12-s − 1.65i·13-s − 0.461·14-s + 0.507·16-s − 1.27i·17-s − 0.771i·18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273854 - 1.16006i\)
\(L(\frac12)\) \(\approx\) \(0.273854 - 1.16006i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 + 2.37iT - 2T^{2} \)
3 \( 1 - 1.27iT - 3T^{2} \)
7 \( 1 + 0.726iT - 7T^{2} \)
11 \( 1 + 0.273T + 11T^{2} \)
13 \( 1 + 5.95iT - 13T^{2} \)
17 \( 1 + 5.27iT - 17T^{2} \)
23 \( 1 + 3.67iT - 23T^{2} \)
29 \( 1 - 2.27T + 29T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 + 8.12iT - 37T^{2} \)
41 \( 1 + 9.43T + 41T^{2} \)
43 \( 1 - 9.81iT - 43T^{2} \)
47 \( 1 + 12.1iT - 47T^{2} \)
53 \( 1 - 5.69iT - 53T^{2} \)
59 \( 1 - 4.20T + 59T^{2} \)
61 \( 1 + 0.103T + 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 - 5.75T + 71T^{2} \)
73 \( 1 - 6.67iT - 73T^{2} \)
79 \( 1 + 3.87T + 79T^{2} \)
83 \( 1 + 0.488iT - 83T^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 - 4.44iT - 97T^{2} \)
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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

−10.36644158055459173611890394190, −10.24650933340457964363552455128, −9.293871947206007969760416534551, −8.321133509363248593547069638526, −7.05275006833341500885039349252, −5.33195578376663913744324793332, −4.49570962617696038617552861540, −3.49703510890838847223918196093, −2.54771212301762541711046823929, −0.77630648674142647658818347821, 1.80531395515927129298793889554, 4.01258524216955506088790076057, 5.00210981434479267325767906747, 6.30599571365374834331701642919, 6.63713441751894087032892949133, 7.59508916265941725592877222876, 8.382554217532235293253610767029, 9.169430507057608980654995886158, 10.21233640645040518357040816073, 11.62737194851000376682928984104

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