Properties

Label 2-475-1.1-c1-0-8
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 2·4-s + 7-s + 9-s + 3·11-s − 4·12-s + 4·13-s + 4·16-s + 3·17-s + 19-s + 2·21-s − 4·27-s − 2·28-s + 6·29-s − 4·31-s + 6·33-s − 2·36-s − 2·37-s + 8·39-s − 6·41-s + 43-s − 6·44-s + 3·47-s + 8·48-s − 6·49-s + 6·51-s − 8·52-s + ⋯
L(s)  = 1  + 1.15·3-s − 4-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 1.15·12-s + 1.10·13-s + 16-s + 0.727·17-s + 0.229·19-s + 0.436·21-s − 0.769·27-s − 0.377·28-s + 1.11·29-s − 0.718·31-s + 1.04·33-s − 1/3·36-s − 0.328·37-s + 1.28·39-s − 0.937·41-s + 0.152·43-s − 0.904·44-s + 0.437·47-s + 1.15·48-s − 6/7·49-s + 0.840·51-s − 1.10·52-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: $\chi_{475} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.824309118\)
\(L(\frac12)\) \(\approx\) \(1.824309118\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−10.91480268940863095584089699098, −9.774569434898319390985970077664, −9.093042453849772655367942297608, −8.415277882855381679327851057927, −7.79259928613141419667093336124, −6.36395749589543487340848898303, −5.15254746234400988386261721371, −3.94480584782122104516168200313, −3.23182893022898332662250292694, −1.43009506463758030263279983978, 1.43009506463758030263279983978, 3.23182893022898332662250292694, 3.94480584782122104516168200313, 5.15254746234400988386261721371, 6.36395749589543487340848898303, 7.79259928613141419667093336124, 8.415277882855381679327851057927, 9.093042453849772655367942297608, 9.774569434898319390985970077664, 10.91480268940863095584089699098

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