Properties

Label 2-2475-5.4-c1-0-42
Degree $2$
Conductor $2475$
Sign $0.447 - 0.894i$
Analytic cond. $19.7629$
Root an. cond. $4.44555$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.70i·2-s − 5.34·4-s + 1.07i·7-s − 9.04i·8-s − 11-s + 4.34i·13-s − 2.92·14-s + 13.8·16-s − 7.75i·17-s − 5.26·19-s − 2.70i·22-s − 2.15i·23-s − 11.7·26-s − 5.75i·28-s + 1.41·29-s + ⋯
L(s)  = 1  + 1.91i·2-s − 2.67·4-s + 0.407i·7-s − 3.19i·8-s − 0.301·11-s + 1.20i·13-s − 0.780·14-s + 3.45·16-s − 1.88i·17-s − 1.20·19-s − 0.577i·22-s − 0.449i·23-s − 2.30·26-s − 1.08i·28-s + 0.263·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2475\)    =    \(3^{2} \cdot 5^{2} \cdot 11\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(19.7629\)
Root analytic conductor: \(4.44555\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2475} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9070606186\)
\(L(\frac12)\) \(\approx\) \(0.9070606186\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70iT - 2T^{2} \)
7 \( 1 - 1.07iT - 7T^{2} \)
13 \( 1 - 4.34iT - 13T^{2} \)
17 \( 1 + 7.75iT - 17T^{2} \)
19 \( 1 + 5.26T + 19T^{2} \)
23 \( 1 + 2.15iT - 23T^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 + 4.68T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 - 9.41T + 41T^{2} \)
43 \( 1 + 7.60iT - 43T^{2} \)
47 \( 1 + 4.68iT - 47T^{2} \)
53 \( 1 - 0.156iT - 53T^{2} \)
59 \( 1 - 6.15T + 59T^{2} \)
61 \( 1 + 4.15T + 61T^{2} \)
67 \( 1 + 8.68iT - 67T^{2} \)
71 \( 1 - 4.68T + 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 - 8.09T + 79T^{2} \)
83 \( 1 + 11.0iT - 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 - 14.6iT - 97T^{2} \)
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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

−9.033393804666639007121663781654, −8.190112826189894619211240229190, −7.27800594060438617648647151169, −6.89730945643482612790418156988, −6.07959538305745469175568880503, −5.30892231709228951463812696064, −4.62260699065714203854573936568, −3.87265958742958798907124254472, −2.36960266861791469077501644195, −0.38550082024870081196225663818, 0.962477082539655104934440808895, 1.98880679935300798845728088717, 2.91561729531046183238664761078, 3.80701178154715864801962924211, 4.35883714395087768627411125925, 5.40449309743178553647481681833, 6.18381353861574135602713721950, 7.72269076607254566545608202073, 8.273251774845343060782098013221, 8.996308824575314681783105746393

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