Properties

Label 2-1425-1.1-c1-0-10
Degree $2$
Conductor $1425$
Sign $1$
Analytic cond. $11.3786$
Root an. cond. $3.37323$
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  − 2.66·2-s + 3-s + 5.12·4-s − 2.66·6-s + 0.454·7-s − 8.33·8-s + 9-s − 1.45·11-s + 5.12·12-s − 3.33·13-s − 1.21·14-s + 12.0·16-s − 2.66·18-s + 19-s + 0.454·21-s + 3.88·22-s − 6.79·23-s − 8.33·24-s + 8.90·26-s + 27-s + 2.33·28-s + 3·29-s + 8.79·31-s − 15.3·32-s − 1.45·33-s + 5.12·36-s + 10.2·37-s + ⋯
L(s)  = 1  − 1.88·2-s + 0.577·3-s + 2.56·4-s − 1.08·6-s + 0.171·7-s − 2.94·8-s + 0.333·9-s − 0.438·11-s + 1.47·12-s − 0.925·13-s − 0.324·14-s + 3.00·16-s − 0.629·18-s + 0.229·19-s + 0.0992·21-s + 0.827·22-s − 1.41·23-s − 1.70·24-s + 1.74·26-s + 0.192·27-s + 0.440·28-s + 0.557·29-s + 1.57·31-s − 2.71·32-s − 0.253·33-s + 0.853·36-s + 1.68·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(11.3786\)
Root analytic conductor: \(3.37323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1425,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8031940834\)
\(L(\frac12)\) \(\approx\) \(0.8031940834\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
19 \( 1 - T \)
good2 \( 1 + 2.66T + 2T^{2} \)
7 \( 1 - 0.454T + 7T^{2} \)
11 \( 1 + 1.45T + 11T^{2} \)
13 \( 1 + 3.33T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 + 6.79T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 8.79T + 31T^{2} \)
37 \( 1 - 10.2T + 37T^{2} \)
41 \( 1 - 3.97T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 2.42T + 47T^{2} \)
53 \( 1 - 13.6T + 53T^{2} \)
59 \( 1 - 7.54T + 59T^{2} \)
61 \( 1 + 3.90T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + 9.13T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 8.97T + 79T^{2} \)
83 \( 1 + 4.54T + 83T^{2} \)
89 \( 1 + 0.0901T + 89T^{2} \)
97 \( 1 - 12.0T + 97T^{2} \)
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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

−9.609839927894325207105117949222, −8.712532779057635404647211517605, −7.981433715685128735233749861301, −7.60583346852054660816995680653, −6.68840310874937769841187580499, −5.77288139647343232522822230986, −4.32664668896782373582464097133, −2.78556194172172236322488934756, −2.21476187028307508214873968040, −0.817368586730479305433292227350, 0.817368586730479305433292227350, 2.21476187028307508214873968040, 2.78556194172172236322488934756, 4.32664668896782373582464097133, 5.77288139647343232522822230986, 6.68840310874937769841187580499, 7.60583346852054660816995680653, 7.981433715685128735233749861301, 8.712532779057635404647211517605, 9.609839927894325207105117949222

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