Properties

Label 2-1425-1.1-c1-0-16
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  − 2.39·2-s + 3-s + 3.72·4-s − 2.39·6-s + 4.15·7-s − 4.13·8-s + 9-s − 5.71·11-s + 3.72·12-s + 3.79·13-s − 9.93·14-s + 2.44·16-s − 2.66·17-s − 2.39·18-s + 19-s + 4.15·21-s + 13.6·22-s + 8.13·23-s − 4.13·24-s − 9.09·26-s + 27-s + 15.4·28-s + 4.73·29-s − 2.31·31-s + 2.42·32-s − 5.71·33-s + 6.38·34-s + ⋯
L(s)  = 1  − 1.69·2-s + 0.577·3-s + 1.86·4-s − 0.977·6-s + 1.56·7-s − 1.46·8-s + 0.333·9-s − 1.72·11-s + 1.07·12-s + 1.05·13-s − 2.65·14-s + 0.610·16-s − 0.646·17-s − 0.564·18-s + 0.229·19-s + 0.906·21-s + 2.91·22-s + 1.69·23-s − 0.844·24-s − 1.78·26-s + 0.192·27-s + 2.92·28-s + 0.878·29-s − 0.415·31-s + 0.428·32-s − 0.995·33-s + 1.09·34-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

\(L(1)\) \(\approx\) \(1.090397187\)
\(L(\frac12)\) \(\approx\) \(1.090397187\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.39T + 2T^{2} \)
7 \( 1 - 4.15T + 7T^{2} \)
11 \( 1 + 5.71T + 11T^{2} \)
13 \( 1 - 3.79T + 13T^{2} \)
17 \( 1 + 2.66T + 17T^{2} \)
23 \( 1 - 8.13T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 + 2.31T + 31T^{2} \)
37 \( 1 + 4.68T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 1.76T + 43T^{2} \)
47 \( 1 - 7.14T + 47T^{2} \)
53 \( 1 + 3.04T + 53T^{2} \)
59 \( 1 + 0.582T + 59T^{2} \)
61 \( 1 + 9.54T + 61T^{2} \)
67 \( 1 + 1.20T + 67T^{2} \)
71 \( 1 - 1.20T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 5.06T + 79T^{2} \)
83 \( 1 - 1.83T + 83T^{2} \)
89 \( 1 - 3.36T + 89T^{2} \)
97 \( 1 - 0.313T + 97T^{2} \)
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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

−9.212255645485877232424656358630, −8.735670669198710472445147998114, −8.012315527350309643498535492697, −7.66134322626100280387941106439, −6.76104683755449074158998367121, −5.42809281217360993473538606480, −4.53614947214681502907588263069, −2.92190288206840948444097653772, −2.03233052475643602027243928755, −0.987322599412682405163559365636, 0.987322599412682405163559365636, 2.03233052475643602027243928755, 2.92190288206840948444097653772, 4.53614947214681502907588263069, 5.42809281217360993473538606480, 6.76104683755449074158998367121, 7.66134322626100280387941106439, 8.012315527350309643498535492697, 8.735670669198710472445147998114, 9.212255645485877232424656358630

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