Properties

Label 2-475-1.1-c1-0-26
Degree $2$
Conductor $475$
Sign $-1$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 2·7-s − 3·8-s − 3·9-s − 4·11-s + 2·13-s − 2·14-s − 16-s − 4·17-s − 3·18-s + 19-s − 4·22-s + 6·23-s + 2·26-s + 2·28-s − 6·29-s − 4·31-s + 5·32-s − 4·34-s + 3·36-s + 10·37-s + 38-s − 10·41-s − 2·43-s + 4·44-s + 6·46-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.755·7-s − 1.06·8-s − 9-s − 1.20·11-s + 0.554·13-s − 0.534·14-s − 1/4·16-s − 0.970·17-s − 0.707·18-s + 0.229·19-s − 0.852·22-s + 1.25·23-s + 0.392·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.883·32-s − 0.685·34-s + 1/2·36-s + 1.64·37-s + 0.162·38-s − 1.56·41-s − 0.304·43-s + 0.603·44-s + 0.884·46-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 18 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−10.75372555761590942712128571003, −9.497197103514413153205439327666, −8.884496394435335630734933815790, −7.910634270974764100222562199420, −6.56197008088421338178147885329, −5.66703630365227380430433843993, −4.88960236344040300877313120060, −3.56253018952603651968369317882, −2.71685356445346523746983648243, 0, 2.71685356445346523746983648243, 3.56253018952603651968369317882, 4.88960236344040300877313120060, 5.66703630365227380430433843993, 6.56197008088421338178147885329, 7.910634270974764100222562199420, 8.884496394435335630734933815790, 9.497197103514413153205439327666, 10.75372555761590942712128571003

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