Properties

Label 2-1425-1.1-c1-0-8
Degree $2$
Conductor $1425$
Sign $1$
Analytic cond. $11.3786$
Root an. cond. $3.37323$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 6-s + 3·8-s + 9-s + 5·11-s + 12-s + 4·13-s − 16-s − 4·17-s − 18-s − 19-s − 5·22-s − 9·23-s − 3·24-s − 4·26-s − 27-s + 7·29-s + 3·31-s − 5·32-s − 5·33-s + 4·34-s − 36-s − 10·37-s + 38-s − 4·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 1.50·11-s + 0.288·12-s + 1.10·13-s − 1/4·16-s − 0.970·17-s − 0.235·18-s − 0.229·19-s − 1.06·22-s − 1.87·23-s − 0.612·24-s − 0.784·26-s − 0.192·27-s + 1.29·29-s + 0.538·31-s − 0.883·32-s − 0.870·33-s + 0.685·34-s − 1/6·36-s − 1.64·37-s + 0.162·38-s − 0.640·39-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1425,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8415530260\)
\(L(\frac12)\) \(\approx\) \(0.8415530260\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 + 15 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−9.561441370849687579716007625047, −8.559969876361858219788163822976, −8.399919956373444667684985492191, −6.98515498693075208426360294141, −6.44029770462908743404015128774, −5.47270146644958024862708785711, −4.24850220460461882940935059602, −3.87301531439376933769109060020, −1.89916774879802682118008269843, −0.794038070961207158407211508808, 0.794038070961207158407211508808, 1.89916774879802682118008269843, 3.87301531439376933769109060020, 4.24850220460461882940935059602, 5.47270146644958024862708785711, 6.44029770462908743404015128774, 6.98515498693075208426360294141, 8.399919956373444667684985492191, 8.559969876361858219788163822976, 9.561441370849687579716007625047

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