Properties

Label 2-2475-1.1-c1-0-62
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  − 2.17·2-s + 2.70·4-s + 3.70·7-s − 1.53·8-s − 11-s − 1.70·13-s − 8.04·14-s − 2.07·16-s + 6.04·17-s − 3.07·19-s + 2.17·22-s − 4·23-s + 3.70·26-s + 10.0·28-s − 5.26·29-s − 6.34·31-s + 7.58·32-s − 13.1·34-s − 3.41·37-s + 6.68·38-s − 9.57·41-s − 3.12·43-s − 2.70·44-s + 8.68·46-s − 2.73·47-s + 6.75·49-s − 4.63·52-s + ⋯
L(s)  = 1  − 1.53·2-s + 1.35·4-s + 1.40·7-s − 0.544·8-s − 0.301·11-s − 0.474·13-s − 2.15·14-s − 0.519·16-s + 1.46·17-s − 0.706·19-s + 0.462·22-s − 0.834·23-s + 0.727·26-s + 1.89·28-s − 0.977·29-s − 1.13·31-s + 1.34·32-s − 2.25·34-s − 0.562·37-s + 1.08·38-s − 1.49·41-s − 0.476·43-s − 0.408·44-s + 1.27·46-s − 0.399·47-s + 0.965·49-s − 0.642·52-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: no
Arithmetic: yes
Character: Trivial
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 \( 1 + 2.17T + 2T^{2} \)
7 \( 1 - 3.70T + 7T^{2} \)
13 \( 1 + 1.70T + 13T^{2} \)
17 \( 1 - 6.04T + 17T^{2} \)
19 \( 1 + 3.07T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 5.26T + 29T^{2} \)
31 \( 1 + 6.34T + 31T^{2} \)
37 \( 1 + 3.41T + 37T^{2} \)
41 \( 1 + 9.57T + 41T^{2} \)
43 \( 1 + 3.12T + 43T^{2} \)
47 \( 1 + 2.73T + 47T^{2} \)
53 \( 1 + 13.7T + 53T^{2} \)
59 \( 1 - 3.60T + 59T^{2} \)
61 \( 1 + 14.6T + 61T^{2} \)
67 \( 1 - 1.84T + 67T^{2} \)
71 \( 1 - 7.23T + 71T^{2} \)
73 \( 1 - 6.38T + 73T^{2} \)
79 \( 1 + 7.44T + 79T^{2} \)
83 \( 1 - 7.86T + 83T^{2} \)
89 \( 1 - 5.02T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−8.468011384803371787375726442353, −7.81920611287335545793371627043, −7.57709793800404457020133761072, −6.51545925301593210653588182068, −5.40745465744103171265371141592, −4.72399889439661067093111268191, −3.48270238010063447432081845998, −2.04784110836699239653169217130, −1.50847258840660794552701925734, 0, 1.50847258840660794552701925734, 2.04784110836699239653169217130, 3.48270238010063447432081845998, 4.72399889439661067093111268191, 5.40745465744103171265371141592, 6.51545925301593210653588182068, 7.57709793800404457020133761072, 7.81920611287335545793371627043, 8.468011384803371787375726442353

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