Properties

Label 2-1445-1.1-c1-0-54
Degree $2$
Conductor $1445$
Sign $-1$
Analytic cond. $11.5383$
Root an. cond. $3.39681$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.80·2-s + 0.687·3-s + 1.26·4-s + 5-s − 1.24·6-s − 4.34·7-s + 1.33·8-s − 2.52·9-s − 1.80·10-s − 0.0525·11-s + 0.867·12-s + 3.02·13-s + 7.84·14-s + 0.687·15-s − 4.93·16-s + 4.56·18-s + 7.82·19-s + 1.26·20-s − 2.98·21-s + 0.0948·22-s + 1.04·23-s + 0.918·24-s + 25-s − 5.46·26-s − 3.80·27-s − 5.47·28-s − 0.420·29-s + ⋯
L(s)  = 1  − 1.27·2-s + 0.397·3-s + 0.630·4-s + 0.447·5-s − 0.507·6-s − 1.64·7-s + 0.471·8-s − 0.842·9-s − 0.571·10-s − 0.0158·11-s + 0.250·12-s + 0.839·13-s + 2.09·14-s + 0.177·15-s − 1.23·16-s + 1.07·18-s + 1.79·19-s + 0.281·20-s − 0.651·21-s + 0.0202·22-s + 0.217·23-s + 0.187·24-s + 0.200·25-s − 1.07·26-s − 0.731·27-s − 1.03·28-s − 0.0781·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1445\)    =    \(5 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(11.5383\)
Root analytic conductor: \(3.39681\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1445,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
17 \( 1 \)
good2 \( 1 + 1.80T + 2T^{2} \)
3 \( 1 - 0.687T + 3T^{2} \)
7 \( 1 + 4.34T + 7T^{2} \)
11 \( 1 + 0.0525T + 11T^{2} \)
13 \( 1 - 3.02T + 13T^{2} \)
19 \( 1 - 7.82T + 19T^{2} \)
23 \( 1 - 1.04T + 23T^{2} \)
29 \( 1 + 0.420T + 29T^{2} \)
31 \( 1 - 1.38T + 31T^{2} \)
37 \( 1 - 0.336T + 37T^{2} \)
41 \( 1 + 6.59T + 41T^{2} \)
43 \( 1 + 9.99T + 43T^{2} \)
47 \( 1 + 6.13T + 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + 5.97T + 61T^{2} \)
67 \( 1 + 0.916T + 67T^{2} \)
71 \( 1 - 4.17T + 71T^{2} \)
73 \( 1 + 5.39T + 73T^{2} \)
79 \( 1 + 9.98T + 79T^{2} \)
83 \( 1 - 6.53T + 83T^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + 19.2T + 97T^{2} \)
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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

−9.280590577751449854053512809153, −8.526635624653893695280499207482, −7.76545023849077954126594307073, −6.79380846295355365628262463832, −6.13531818041269400846136361854, −5.09677864286966440876310271570, −3.49159486780503973117175528517, −2.90621645228161039720524523040, −1.43614997498796148765797655920, 0, 1.43614997498796148765797655920, 2.90621645228161039720524523040, 3.49159486780503973117175528517, 5.09677864286966440876310271570, 6.13531818041269400846136361854, 6.79380846295355365628262463832, 7.76545023849077954126594307073, 8.526635624653893695280499207482, 9.280590577751449854053512809153

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