Properties

Label 2-2475-1.1-c3-0-56
Degree $2$
Conductor $2475$
Sign $1$
Analytic cond. $146.029$
Root an. cond. $12.0842$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.56·2-s − 1.43·4-s − 6.24·7-s + 24.1·8-s + 11·11-s + 49.1·13-s + 16·14-s − 50.4·16-s + 82.7·17-s − 130.·19-s − 28.1·22-s − 185.·23-s − 125.·26-s + 8.98·28-s + 8.90·29-s + 5.26·31-s − 64.2·32-s − 211.·34-s + 416.·37-s + 333.·38-s + 298.·41-s + 513.·43-s − 15.8·44-s + 475.·46-s + 557.·47-s − 303.·49-s − 70.6·52-s + ⋯
L(s)  = 1  − 0.905·2-s − 0.179·4-s − 0.337·7-s + 1.06·8-s + 0.301·11-s + 1.04·13-s + 0.305·14-s − 0.787·16-s + 1.17·17-s − 1.57·19-s − 0.273·22-s − 1.68·23-s − 0.949·26-s + 0.0606·28-s + 0.0570·29-s + 0.0304·31-s − 0.354·32-s − 1.06·34-s + 1.85·37-s + 1.42·38-s + 1.13·41-s + 1.82·43-s − 0.0542·44-s + 1.52·46-s + 1.72·47-s − 0.886·49-s − 0.188·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(146.029\)
Root analytic conductor: \(12.0842\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.068835911\)
\(L(\frac12)\) \(\approx\) \(1.068835911\)
\(L(\frac{5}{2})\) 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 \( 1 - 11T \)
good2 \( 1 + 2.56T + 8T^{2} \)
7 \( 1 + 6.24T + 343T^{2} \)
13 \( 1 - 49.1T + 2.19e3T^{2} \)
17 \( 1 - 82.7T + 4.91e3T^{2} \)
19 \( 1 + 130.T + 6.85e3T^{2} \)
23 \( 1 + 185.T + 1.21e4T^{2} \)
29 \( 1 - 8.90T + 2.43e4T^{2} \)
31 \( 1 - 5.26T + 2.97e4T^{2} \)
37 \( 1 - 416.T + 5.06e4T^{2} \)
41 \( 1 - 298.T + 6.89e4T^{2} \)
43 \( 1 - 513.T + 7.95e4T^{2} \)
47 \( 1 - 557.T + 1.03e5T^{2} \)
53 \( 1 + 168.T + 1.48e5T^{2} \)
59 \( 1 + 618.T + 2.05e5T^{2} \)
61 \( 1 - 786.T + 2.26e5T^{2} \)
67 \( 1 - 339.T + 3.00e5T^{2} \)
71 \( 1 + 1.12e3T + 3.57e5T^{2} \)
73 \( 1 - 123.T + 3.89e5T^{2} \)
79 \( 1 + 309.T + 4.93e5T^{2} \)
83 \( 1 + 1.02e3T + 5.71e5T^{2} \)
89 \( 1 - 141.T + 7.04e5T^{2} \)
97 \( 1 + 798.T + 9.12e5T^{2} \)
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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

−8.534835576698951568378966764115, −8.036335358098949102577723822730, −7.32649292440202265912358086155, −6.17731322396329652214675722492, −5.78112780741648618080260284891, −4.25544151534436078484081468657, −4.01160124147327670174227973821, −2.59073616851752872733235022870, −1.45701157626238585353517137963, −0.56781221964193608466711584279, 0.56781221964193608466711584279, 1.45701157626238585353517137963, 2.59073616851752872733235022870, 4.01160124147327670174227973821, 4.25544151534436078484081468657, 5.78112780741648618080260284891, 6.17731322396329652214675722492, 7.32649292440202265912358086155, 8.036335358098949102577723822730, 8.534835576698951568378966764115

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