Properties

Label 2-2475-1.1-c1-0-47
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·2-s + 3.82·4-s + 2·7-s + 4.41·8-s − 11-s + 1.17·13-s + 4.82·14-s + 2.99·16-s + 6.82·17-s − 2.41·22-s − 2.82·23-s + 2.82·26-s + 7.65·28-s + 3.65·29-s − 1.58·32-s + 16.4·34-s + 7.65·37-s − 6·41-s + 6·43-s − 3.82·44-s − 6.82·46-s + 2.82·47-s − 3·49-s + 4.48·52-s + 11.6·53-s + 8.82·56-s + 8.82·58-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.91·4-s + 0.755·7-s + 1.56·8-s − 0.301·11-s + 0.324·13-s + 1.29·14-s + 0.749·16-s + 1.65·17-s − 0.514·22-s − 0.589·23-s + 0.554·26-s + 1.44·28-s + 0.679·29-s − 0.280·32-s + 2.82·34-s + 1.25·37-s − 0.937·41-s + 0.914·43-s − 0.577·44-s − 1.00·46-s + 0.412·47-s − 0.428·49-s + 0.621·52-s + 1.60·53-s + 1.17·56-s + 1.15·58-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: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.699684376\)
\(L(\frac12)\) \(\approx\) \(5.699684376\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 7T^{2} \)
13 \( 1 - 1.17T + 13T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 1.17T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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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

−8.775792573652341552029370576849, −7.84436201939499780430693494726, −7.32837030396739543953959643528, −6.19136387220385288591921419836, −5.71026623909183260703198512596, −4.92272567370070635036782669109, −4.23103054301320907439201112597, −3.36029176631802819053516109770, −2.52939341252480033016628415450, −1.35717830261851675732069691262, 1.35717830261851675732069691262, 2.52939341252480033016628415450, 3.36029176631802819053516109770, 4.23103054301320907439201112597, 4.92272567370070635036782669109, 5.71026623909183260703198512596, 6.19136387220385288591921419836, 7.32837030396739543953959643528, 7.84436201939499780430693494726, 8.775792573652341552029370576849

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