Properties

Label 2-2475-1.1-c1-0-44
Degree $2$
Conductor $2475$
Sign $-1$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.41·2-s + 3.82·4-s − 0.828·7-s − 4.41·8-s + 11-s + 5.65·13-s + 1.99·14-s + 2.99·16-s − 1.17·17-s − 6.82·19-s − 2.41·22-s − 4·23-s − 13.6·26-s − 3.17·28-s + 4.82·29-s + 1.58·32-s + 2.82·34-s − 11.6·37-s + 16.4·38-s − 4.82·41-s + 8.82·43-s + 3.82·44-s + 9.65·46-s − 4·47-s − 6.31·49-s + 21.6·52-s + 9.31·53-s + ⋯
L(s)  = 1  − 1.70·2-s + 1.91·4-s − 0.313·7-s − 1.56·8-s + 0.301·11-s + 1.56·13-s + 0.534·14-s + 0.749·16-s − 0.284·17-s − 1.56·19-s − 0.514·22-s − 0.834·23-s − 2.67·26-s − 0.599·28-s + 0.896·29-s + 0.280·32-s + 0.485·34-s − 1.91·37-s + 2.67·38-s − 0.754·41-s + 1.34·43-s + 0.577·44-s + 1.42·46-s − 0.583·47-s − 0.901·49-s + 3.00·52-s + 1.27·53-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: no
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 \( 1 + 2.41T + 2T^{2} \)
7 \( 1 + 0.828T + 7T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
19 \( 1 + 6.82T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 4.82T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 11.6T + 37T^{2} \)
41 \( 1 + 4.82T + 41T^{2} \)
43 \( 1 - 8.82T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 - 9.31T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 5.65T + 67T^{2} \)
71 \( 1 + 2.34T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 8.48T + 79T^{2} \)
83 \( 1 + 10T + 83T^{2} \)
89 \( 1 + 3.65T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−8.586821501530456464839783061432, −8.193105262987933401903597802500, −7.09710671797099572276068316357, −6.48826185665042504057062220566, −5.90661617028775874847721765252, −4.41518627320243611006829144466, −3.44384537625838116142222658420, −2.20812933505978642288681380919, −1.32696562717415949082235892683, 0, 1.32696562717415949082235892683, 2.20812933505978642288681380919, 3.44384537625838116142222658420, 4.41518627320243611006829144466, 5.90661617028775874847721765252, 6.48826185665042504057062220566, 7.09710671797099572276068316357, 8.193105262987933401903597802500, 8.586821501530456464839783061432

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