Properties

Label 2-1425-1.1-c1-0-40
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  + 2.14·2-s + 3-s + 2.60·4-s + 2.14·6-s + 2.74·7-s + 1.29·8-s + 9-s − 3.74·11-s + 2.60·12-s + 6.29·13-s + 5.89·14-s − 2.43·16-s + 2.14·18-s + 19-s + 2.74·21-s − 8.03·22-s + 0.543·23-s + 1.29·24-s + 13.4·26-s + 27-s + 7.14·28-s + 3·29-s + 1.45·31-s − 7.80·32-s − 3.74·33-s + 2.60·36-s + 5.20·37-s + ⋯
L(s)  = 1  + 1.51·2-s + 0.577·3-s + 1.30·4-s + 0.875·6-s + 1.03·7-s + 0.456·8-s + 0.333·9-s − 1.12·11-s + 0.750·12-s + 1.74·13-s + 1.57·14-s − 0.608·16-s + 0.505·18-s + 0.229·19-s + 0.599·21-s − 1.71·22-s + 0.113·23-s + 0.263·24-s + 2.64·26-s + 0.192·27-s + 1.35·28-s + 0.557·29-s + 0.261·31-s − 1.37·32-s − 0.652·33-s + 0.433·36-s + 0.855·37-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

\(L(1)\) \(\approx\) \(5.096504289\)
\(L(\frac12)\) \(\approx\) \(5.096504289\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.14T + 2T^{2} \)
7 \( 1 - 2.74T + 7T^{2} \)
11 \( 1 + 3.74T + 11T^{2} \)
13 \( 1 - 6.29T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
23 \( 1 - 0.543T + 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 1.45T + 31T^{2} \)
37 \( 1 - 5.20T + 37T^{2} \)
41 \( 1 + 12.5T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 + 5.58T + 53T^{2} \)
59 \( 1 - 5.25T + 59T^{2} \)
61 \( 1 + 8.49T + 61T^{2} \)
67 \( 1 + 2.83T + 67T^{2} \)
71 \( 1 - 7.83T + 71T^{2} \)
73 \( 1 - 7.58T + 73T^{2} \)
79 \( 1 + 7.52T + 79T^{2} \)
83 \( 1 + 2.25T + 83T^{2} \)
89 \( 1 - 4.49T + 89T^{2} \)
97 \( 1 + 16.8T + 97T^{2} \)
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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

−9.532140065828462688864815531328, −8.376242849134875061007593236510, −8.080166020208417316138290174897, −6.89144635828577464043125168307, −6.01854000434960821990958855695, −5.17071929849830422259778729572, −4.50086211635991439638389892579, −3.55509853241516110056284173074, −2.75590044579045009746241961286, −1.57432478050364414297587662814, 1.57432478050364414297587662814, 2.75590044579045009746241961286, 3.55509853241516110056284173074, 4.50086211635991439638389892579, 5.17071929849830422259778729572, 6.01854000434960821990958855695, 6.89144635828577464043125168307, 8.080166020208417316138290174897, 8.376242849134875061007593236510, 9.532140065828462688864815531328

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