Properties

Label 2-1425-1.1-c1-0-44
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  + 2·2-s + 3-s + 2·4-s + 2·6-s + 5·7-s + 9-s + 11-s + 2·12-s − 2·13-s + 10·14-s − 4·16-s + 17-s + 2·18-s − 19-s + 5·21-s + 2·22-s + 4·23-s − 4·26-s + 27-s + 10·28-s − 2·29-s − 6·31-s − 8·32-s + 33-s + 2·34-s + 2·36-s − 2·38-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 4-s + 0.816·6-s + 1.88·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s − 0.554·13-s + 2.67·14-s − 16-s + 0.242·17-s + 0.471·18-s − 0.229·19-s + 1.09·21-s + 0.426·22-s + 0.834·23-s − 0.784·26-s + 0.192·27-s + 1.88·28-s − 0.371·29-s − 1.07·31-s − 1.41·32-s + 0.174·33-s + 0.342·34-s + 1/3·36-s − 0.324·38-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

\(L(1)\) \(\approx\) \(4.968994303\)
\(L(\frac12)\) \(\approx\) \(4.968994303\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
7 \( 1 - 5 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−9.341053920747992022463934553266, −8.723409361965311529712840067335, −7.74191106110524052069368926798, −7.15886697828319609358757774448, −5.94366419557132992161838079855, −5.07072622007111059188732569210, −4.55783516638495121923212657411, −3.69330683704517818549980694450, −2.57494569661476323183620903482, −1.60475124369086001622436622243, 1.60475124369086001622436622243, 2.57494569661476323183620903482, 3.69330683704517818549980694450, 4.55783516638495121923212657411, 5.07072622007111059188732569210, 5.94366419557132992161838079855, 7.15886697828319609358757774448, 7.74191106110524052069368926798, 8.723409361965311529712840067335, 9.341053920747992022463934553266

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