Properties

Label 2-475-5.4-c1-0-6
Degree $2$
Conductor $475$
Sign $-0.447 - 0.894i$
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  − 0.311i·2-s + 2.90i·3-s + 1.90·4-s + 0.903·6-s + 4.42i·7-s − 1.21i·8-s − 5.42·9-s − 2.62·11-s + 5.52i·12-s + 0.474i·13-s + 1.37·14-s + 3.42·16-s − 5.05i·17-s + 1.68i·18-s + 19-s + ⋯
L(s)  = 1  − 0.219i·2-s + 1.67i·3-s + 0.951·4-s + 0.368·6-s + 1.67i·7-s − 0.429i·8-s − 1.80·9-s − 0.790·11-s + 1.59i·12-s + 0.131i·13-s + 0.368·14-s + 0.857·16-s − 1.22i·17-s + 0.398i·18-s + 0.229·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 475,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.843435 + 1.36470i\)
\(L(\frac12)\) \(\approx\) \(0.843435 + 1.36470i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.311iT - 2T^{2} \)
3 \( 1 - 2.90iT - 3T^{2} \)
7 \( 1 - 4.42iT - 7T^{2} \)
11 \( 1 + 2.62T + 11T^{2} \)
13 \( 1 - 0.474iT - 13T^{2} \)
17 \( 1 + 5.05iT - 17T^{2} \)
23 \( 1 + 1.37iT - 23T^{2} \)
29 \( 1 - 7.80T + 29T^{2} \)
31 \( 1 - 1.24T + 31T^{2} \)
37 \( 1 + 4.47iT - 37T^{2} \)
41 \( 1 + 5.05T + 41T^{2} \)
43 \( 1 - 12.0iT - 43T^{2} \)
47 \( 1 - 4.42iT - 47T^{2} \)
53 \( 1 - 7.52iT - 53T^{2} \)
59 \( 1 - 2.19T + 59T^{2} \)
61 \( 1 - 3.67T + 61T^{2} \)
67 \( 1 - 1.65iT - 67T^{2} \)
71 \( 1 - 7.61T + 71T^{2} \)
73 \( 1 + 3.80iT - 73T^{2} \)
79 \( 1 - 13.4T + 79T^{2} \)
83 \( 1 + 10.6iT - 83T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + 17.8iT - 97T^{2} \)
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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

−11.25991276298869515461305602778, −10.38248856497458449943756323914, −9.639639164085888516717912494550, −8.905978329009433873800971900938, −7.895018504508552980281590541580, −6.39036593036123170847674868646, −5.45141472762195219990169313067, −4.70864639395934645839245544882, −3.10221209064181506472385403439, −2.54237713377928908844146120341, 0.996035649149581764977218683675, 2.17654260545884217218747931326, 3.53717722217027165533086682704, 5.33877303064088435315633766897, 6.63474283901837241344896074599, 6.85114670027400913979392584121, 7.926374746338273658165290183645, 8.189875333322694638206566343079, 10.23806763669370332366678346773, 10.71238600759131251005371104697

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