Properties

Label 2-2475-1.1-c1-0-6
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  − 1.61·2-s + 0.618·4-s − 2.85·7-s + 2.23·8-s − 11-s + 6.23·13-s + 4.61·14-s − 4.85·16-s + 0.618·17-s − 6.70·19-s + 1.61·22-s + 4.09·23-s − 10.0·26-s − 1.76·28-s + 1.38·29-s − 3·31-s + 3.38·32-s − 1.00·34-s + 10.2·37-s + 10.8·38-s + 3·41-s − 6·43-s − 0.618·44-s − 6.61·46-s − 11.9·47-s + 1.14·49-s + 3.85·52-s + ⋯
L(s)  = 1  − 1.14·2-s + 0.309·4-s − 1.07·7-s + 0.790·8-s − 0.301·11-s + 1.72·13-s + 1.23·14-s − 1.21·16-s + 0.149·17-s − 1.53·19-s + 0.344·22-s + 0.852·23-s − 1.97·26-s − 0.333·28-s + 0.256·29-s − 0.538·31-s + 0.597·32-s − 0.171·34-s + 1.68·37-s + 1.76·38-s + 0.468·41-s − 0.914·43-s − 0.0931·44-s − 0.975·46-s − 1.74·47-s + 0.163·49-s + 0.534·52-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

\(L(1)\) \(\approx\) \(0.6802251722\)
\(L(\frac12)\) \(\approx\) \(0.6802251722\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.61T + 2T^{2} \)
7 \( 1 + 2.85T + 7T^{2} \)
13 \( 1 - 6.23T + 13T^{2} \)
17 \( 1 - 0.618T + 17T^{2} \)
19 \( 1 + 6.70T + 19T^{2} \)
23 \( 1 - 4.09T + 23T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 11.9T + 47T^{2} \)
53 \( 1 + 9.32T + 53T^{2} \)
59 \( 1 + 0.527T + 59T^{2} \)
61 \( 1 - 0.0901T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 8.18T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 - 5.85T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 6.90T + 89T^{2} \)
97 \( 1 - 1.61T + 97T^{2} \)
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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

−8.941659373707782851068029587110, −8.323491105795828004422478385486, −7.68866108884944240775195028373, −6.45817892164130181003547533687, −6.35320534499018699352081772823, −4.97427209371664079860068142091, −3.98800753877066793284880343359, −3.11298100691947915189548158533, −1.81765316420836059210672361330, −0.62962301905626252087556351454, 0.62962301905626252087556351454, 1.81765316420836059210672361330, 3.11298100691947915189548158533, 3.98800753877066793284880343359, 4.97427209371664079860068142091, 6.35320534499018699352081772823, 6.45817892164130181003547533687, 7.68866108884944240775195028373, 8.323491105795828004422478385486, 8.941659373707782851068029587110

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