Properties

Label 2-475-1.1-c1-0-0
Degree $2$
Conductor $475$
Sign $1$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.82·2-s − 2.31·3-s + 1.32·4-s + 4.21·6-s − 1.45·7-s + 1.23·8-s + 2.35·9-s − 3.89·11-s − 3.05·12-s − 3.05·13-s + 2.64·14-s − 4.89·16-s − 3.92·17-s − 4.29·18-s − 19-s + 3.35·21-s + 7.10·22-s + 5.37·23-s − 2.86·24-s + 5.57·26-s + 1.48·27-s − 1.91·28-s + 6·29-s − 8.43·31-s + 6.45·32-s + 9.01·33-s + 7.14·34-s + ⋯
L(s)  = 1  − 1.28·2-s − 1.33·3-s + 0.660·4-s + 1.72·6-s − 0.548·7-s + 0.437·8-s + 0.785·9-s − 1.17·11-s − 0.883·12-s − 0.848·13-s + 0.706·14-s − 1.22·16-s − 0.951·17-s − 1.01·18-s − 0.229·19-s + 0.732·21-s + 1.51·22-s + 1.12·23-s − 0.584·24-s + 1.09·26-s + 0.286·27-s − 0.362·28-s + 1.11·29-s − 1.51·31-s + 1.14·32-s + 1.56·33-s + 1.22·34-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.2380172070\)
\(L(\frac12)\) \(\approx\) \(0.2380172070\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.82T + 2T^{2} \)
3 \( 1 + 2.31T + 3T^{2} \)
7 \( 1 + 1.45T + 7T^{2} \)
11 \( 1 + 3.89T + 11T^{2} \)
13 \( 1 + 3.05T + 13T^{2} \)
17 \( 1 + 3.92T + 17T^{2} \)
23 \( 1 - 5.37T + 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8.43T + 31T^{2} \)
37 \( 1 - 5.95T + 37T^{2} \)
41 \( 1 - 10.4T + 41T^{2} \)
43 \( 1 - 1.45T + 43T^{2} \)
47 \( 1 + 4.90T + 47T^{2} \)
53 \( 1 - 4.23T + 53T^{2} \)
59 \( 1 - 3.35T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 9.84T + 67T^{2} \)
71 \( 1 - 8.64T + 71T^{2} \)
73 \( 1 - 2.43T + 73T^{2} \)
79 \( 1 + 12.4T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 3.05T + 97T^{2} \)
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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

−10.92360812944661267445702362802, −10.14535627485752226750806859598, −9.404337651361221468961072749960, −8.386922451235082410427732340928, −7.32485420807537410536770849765, −6.62092651503056810280681062615, −5.41661366155861086766703204900, −4.55690299393144658623841334770, −2.49386011183497425266052928076, −0.55456626669807145697844161379, 0.55456626669807145697844161379, 2.49386011183497425266052928076, 4.55690299393144658623841334770, 5.41661366155861086766703204900, 6.62092651503056810280681062615, 7.32485420807537410536770849765, 8.386922451235082410427732340928, 9.404337651361221468961072749960, 10.14535627485752226750806859598, 10.92360812944661267445702362802

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