Properties

Label 2-2475-1.1-c1-0-67
Degree $2$
Conductor $2475$
Sign $-1$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
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 − 11-s + 2·13-s + 2·14-s − 16-s − 2·17-s − 6·19-s − 22-s − 4·23-s + 2·26-s − 2·28-s − 6·29-s + 4·31-s + 5·32-s − 2·34-s + 6·37-s − 6·38-s − 10·41-s − 6·43-s + 44-s − 4·46-s + 8·47-s − 3·49-s − 2·52-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s − 0.301·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.213·22-s − 0.834·23-s + 0.392·26-s − 0.377·28-s − 1.11·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s + 0.986·37-s − 0.973·38-s − 1.56·41-s − 0.914·43-s + 0.150·44-s − 0.589·46-s + 1.16·47-s − 3/7·49-s − 0.277·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $-1$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2475} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 + T \)
good2 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.426508541426244909616098012576, −8.037404731553225089340453872858, −6.81603726027801660997015654358, −6.02983209069794989603640383676, −5.32946274439723089843787755675, −4.42806580035514350183447341926, −3.98159649274510364542505380476, −2.82049928275954817840186820477, −1.70002773304928331393197381104, 0, 1.70002773304928331393197381104, 2.82049928275954817840186820477, 3.98159649274510364542505380476, 4.42806580035514350183447341926, 5.32946274439723089843787755675, 6.02983209069794989603640383676, 6.81603726027801660997015654358, 8.037404731553225089340453872858, 8.426508541426244909616098012576

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