Properties

Label 2-475-475.341-c0-0-0
Degree $2$
Conductor $475$
Sign $-0.187 - 0.982i$
Analytic cond. $0.237055$
Root an. cond. $0.486883$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s − 1.61·7-s + (−0.809 + 0.587i)9-s + (1.30 + 0.951i)11-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)17-s + (0.309 − 0.951i)19-s + (−0.809 − 0.587i)20-s + (1.30 + 0.951i)23-s + (0.309 − 0.951i)25-s + (−0.500 − 1.53i)28-s + (1.30 − 0.951i)35-s + (−0.809 − 0.587i)36-s + 0.618·43-s + (−0.499 + 1.53i)44-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)4-s + (−0.809 + 0.587i)5-s − 1.61·7-s + (−0.809 + 0.587i)9-s + (1.30 + 0.951i)11-s + (−0.809 + 0.587i)16-s + (0.190 − 0.587i)17-s + (0.309 − 0.951i)19-s + (−0.809 − 0.587i)20-s + (1.30 + 0.951i)23-s + (0.309 − 0.951i)25-s + (−0.500 − 1.53i)28-s + (1.30 − 0.951i)35-s + (−0.809 − 0.587i)36-s + 0.618·43-s + (−0.499 + 1.53i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.187 - 0.982i$
Analytic conductor: \(0.237055\)
Root analytic conductor: \(0.486883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :0),\ -0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6601589846\)
\(L(\frac12)\) \(\approx\) \(0.6601589846\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + 1.61T + T^{2} \)
11 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 + (-0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−11.57098765060952982556298482045, −10.82485535285528869782389645456, −9.467055822960684928197943573349, −8.938365329466624269561752040084, −7.50875694392123445162509218525, −7.09503056414150789237536816268, −6.22413521839050327094082296587, −4.51293651688591239879326387800, −3.35540762624933005372024145887, −2.77778598257329082561604699066, 0.872287529666106770604159254580, 3.11529654277132639688758752091, 3.95777207454663036647427554552, 5.58527622991177046775248432152, 6.26518154266451488968075602242, 7.02039870330467664178975716433, 8.650388957596350252990442803839, 9.082053096475251744386313063417, 10.03522390801117895182415332128, 11.03471040038214667143058997547

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