Properties

Label 2-1425-1.1-c1-0-50
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  + 2.73·2-s + 3-s + 5.48·4-s + 2.73·6-s + 2.95·7-s + 9.52·8-s + 9-s − 4.70·11-s + 5.48·12-s − 3.69·13-s + 8.08·14-s + 15.0·16-s − 4.86·17-s + 2.73·18-s + 19-s + 2.95·21-s − 12.8·22-s − 2.38·23-s + 9.52·24-s − 10.1·26-s + 27-s + 16.1·28-s − 6.56·29-s + 1.78·31-s + 22.2·32-s − 4.70·33-s − 13.3·34-s + ⋯
L(s)  = 1  + 1.93·2-s + 0.577·3-s + 2.74·4-s + 1.11·6-s + 1.11·7-s + 3.36·8-s + 0.333·9-s − 1.41·11-s + 1.58·12-s − 1.02·13-s + 2.15·14-s + 3.77·16-s − 1.18·17-s + 0.644·18-s + 0.229·19-s + 0.644·21-s − 2.74·22-s − 0.497·23-s + 1.94·24-s − 1.98·26-s + 0.192·27-s + 3.06·28-s − 1.21·29-s + 0.319·31-s + 3.92·32-s − 0.819·33-s − 2.28·34-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

\(L(1)\) \(\approx\) \(6.886654022\)
\(L(\frac12)\) \(\approx\) \(6.886654022\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.73T + 2T^{2} \)
7 \( 1 - 2.95T + 7T^{2} \)
11 \( 1 + 4.70T + 11T^{2} \)
13 \( 1 + 3.69T + 13T^{2} \)
17 \( 1 + 4.86T + 17T^{2} \)
23 \( 1 + 2.38T + 23T^{2} \)
29 \( 1 + 6.56T + 29T^{2} \)
31 \( 1 - 1.78T + 31T^{2} \)
37 \( 1 - 3.73T + 37T^{2} \)
41 \( 1 - 2.39T + 41T^{2} \)
43 \( 1 + 8.82T + 43T^{2} \)
47 \( 1 - 1.48T + 47T^{2} \)
53 \( 1 + 1.31T + 53T^{2} \)
59 \( 1 + 2.04T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 - 5.51T + 67T^{2} \)
71 \( 1 + 5.51T + 71T^{2} \)
73 \( 1 - 5.57T + 73T^{2} \)
79 \( 1 + 5.18T + 79T^{2} \)
83 \( 1 - 6.83T + 83T^{2} \)
89 \( 1 - 7.13T + 89T^{2} \)
97 \( 1 - 9.88T + 97T^{2} \)
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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

−9.803322833445578945691050233880, −8.344085693270413264528472251222, −7.65596063618950339624242378027, −7.08085166464169199861369601515, −5.94913722610394518212958559346, −4.99071125940161123458352558935, −4.69072910300643808308177174760, −3.61395822346537152470895871587, −2.48201975974414638158174047922, −2.01239289406869129022996892180, 2.01239289406869129022996892180, 2.48201975974414638158174047922, 3.61395822346537152470895871587, 4.69072910300643808308177174760, 4.99071125940161123458352558935, 5.94913722610394518212958559346, 7.08085166464169199861369601515, 7.65596063618950339624242378027, 8.344085693270413264528472251222, 9.803322833445578945691050233880

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