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Properties

Label 2-475-5.4-c1-0-9
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  + 1.27i·2-s + 1.65i·3-s + 0.377·4-s − 2.10·6-s + 3.65i·7-s + 3.02i·8-s + 0.273·9-s + 2.65·11-s + 0.622i·12-s − 6.13i·13-s − 4.65·14-s − 3.10·16-s + 2.34i·17-s + 0.348i·18-s − 19-s + ⋯
L(s)  = 1  + 0.900i·2-s + 0.953i·3-s + 0.188·4-s − 0.858·6-s + 1.37i·7-s + 1.07i·8-s + 0.0912·9-s + 0.799·11-s + 0.179i·12-s − 1.70i·13-s − 1.24·14-s − 0.775·16-s + 0.569i·17-s + 0.0822i·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

\(L(1)\) \(\approx\) \(0.383936 + 1.62638i\)
\(L(\frac12)\) \(\approx\) \(0.383936 + 1.62638i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.27iT - 2T^{2} \)
3 \( 1 - 1.65iT - 3T^{2} \)
7 \( 1 - 3.65iT - 7T^{2} \)
11 \( 1 - 2.65T + 11T^{2} \)
13 \( 1 + 6.13iT - 13T^{2} \)
17 \( 1 - 2.34iT - 17T^{2} \)
23 \( 1 + 5.48iT - 23T^{2} \)
29 \( 1 + 0.651T + 29T^{2} \)
31 \( 1 + 6.67T + 31T^{2} \)
37 \( 1 + 8.70iT - 37T^{2} \)
41 \( 1 - 1.93T + 41T^{2} \)
43 \( 1 - 2.65iT - 43T^{2} \)
47 \( 1 - 3.71iT - 47T^{2} \)
53 \( 1 + 13.7iT - 53T^{2} \)
59 \( 1 - 7.84T + 59T^{2} \)
61 \( 1 + 1.92T + 61T^{2} \)
67 \( 1 + 4.44iT - 67T^{2} \)
71 \( 1 - 3.54T + 71T^{2} \)
73 \( 1 - 2.48iT - 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 - 5.06T + 89T^{2} \)
97 \( 1 - 3.22iT - 97T^{2} \)
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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

−11.16088688504006662722730404326, −10.53084012501705393318992069755, −9.413540091375989854689213306615, −8.651155709458814762597196302799, −7.85000029251860074406846126566, −6.61484039715692411447055451789, −5.71288062202192544679471992978, −5.08011489686288432605022771968, −3.65493824072221312134780137273, −2.30349300319789999501504063094, 1.13604571833398348087700464347, 1.95462892868371112159918293067, 3.61456565048904532477824036870, 4.39446479170640736799797195350, 6.36475511902536047336481280950, 7.05333417228798679656757557806, 7.49033419502169392513978140275, 9.112298041363744161358540084875, 9.871459799094369748043888131177, 10.83423176797868817533960782823

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