Properties

Label 2-2475-1.1-c1-0-31
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  + 1.21·2-s − 0.525·4-s + 4.90·7-s − 3.06·8-s + 11-s + 4.14·13-s + 5.95·14-s − 2.67·16-s + 5.33·17-s − 5.18·19-s + 1.21·22-s − 4·23-s + 5.03·26-s − 2.57·28-s − 1.80·29-s + 2.62·31-s + 2.88·32-s + 6.47·34-s + 5.80·37-s − 6.29·38-s − 1.80·41-s + 4.90·43-s − 0.525·44-s − 4.85·46-s − 7.05·47-s + 17.0·49-s − 2.17·52-s + ⋯
L(s)  = 1  + 0.858·2-s − 0.262·4-s + 1.85·7-s − 1.08·8-s + 0.301·11-s + 1.15·13-s + 1.59·14-s − 0.668·16-s + 1.29·17-s − 1.18·19-s + 0.258·22-s − 0.834·23-s + 0.987·26-s − 0.486·28-s − 0.335·29-s + 0.470·31-s + 0.510·32-s + 1.11·34-s + 0.954·37-s − 1.02·38-s − 0.282·41-s + 0.747·43-s − 0.0792·44-s − 0.716·46-s − 1.02·47-s + 2.43·49-s − 0.302·52-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: \(0\)
Selberg data: \((2,\ 2475,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.146490612\)
\(L(\frac12)\) \(\approx\) \(3.146490612\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.21T + 2T^{2} \)
7 \( 1 - 4.90T + 7T^{2} \)
13 \( 1 - 4.14T + 13T^{2} \)
17 \( 1 - 5.33T + 17T^{2} \)
19 \( 1 + 5.18T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 1.80T + 29T^{2} \)
31 \( 1 - 2.62T + 31T^{2} \)
37 \( 1 - 5.80T + 37T^{2} \)
41 \( 1 + 1.80T + 41T^{2} \)
43 \( 1 - 4.90T + 43T^{2} \)
47 \( 1 + 7.05T + 47T^{2} \)
53 \( 1 - 7.18T + 53T^{2} \)
59 \( 1 + 1.67T + 59T^{2} \)
61 \( 1 - 0.755T + 61T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 + 0.428T + 71T^{2} \)
73 \( 1 - 12.7T + 73T^{2} \)
79 \( 1 + 6.42T + 79T^{2} \)
83 \( 1 - 2.90T + 83T^{2} \)
89 \( 1 + 0.622T + 89T^{2} \)
97 \( 1 + 2.75T + 97T^{2} \)
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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

−8.611418426049659731156187801170, −8.319440568842977582502777915081, −7.52213522491542998239252820619, −6.23457926262712208796389110711, −5.72652633179915019160833579688, −4.85498820906525652532797636383, −4.22545949719542738448259963100, −3.52725488187448770204150681972, −2.18552927336778953146604969676, −1.08868185482885373761069613152, 1.08868185482885373761069613152, 2.18552927336778953146604969676, 3.52725488187448770204150681972, 4.22545949719542738448259963100, 4.85498820906525652532797636383, 5.72652633179915019160833579688, 6.23457926262712208796389110711, 7.52213522491542998239252820619, 8.319440568842977582502777915081, 8.611418426049659731156187801170

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