Properties

Label 2-1045-1.1-c1-0-52
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s + 5-s − 3·8-s − 3·9-s + 10-s − 11-s + 2·13-s − 16-s − 6·17-s − 3·18-s + 19-s − 20-s − 22-s − 8·23-s + 25-s + 2·26-s − 6·29-s + 4·31-s + 5·32-s − 6·34-s + 3·36-s − 2·37-s + 38-s − 3·40-s − 10·41-s + 4·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s − 9-s + 0.316·10-s − 0.301·11-s + 0.554·13-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.229·19-s − 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s − 1.11·29-s + 0.718·31-s + 0.883·32-s − 1.02·34-s + 1/2·36-s − 0.328·37-s + 0.162·38-s − 0.474·40-s − 1.56·41-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−9.430298773506168656487379834590, −8.680265915687738043015894417363, −8.071416084937120999405606330586, −6.61497057113220838361920165158, −5.92455359343492777337204179724, −5.20480645886684182265145788830, −4.22208505788091945721791822553, −3.26824506410830084509707132514, −2.13131620331563136285075125502, 0, 2.13131620331563136285075125502, 3.26824506410830084509707132514, 4.22208505788091945721791822553, 5.20480645886684182265145788830, 5.92455359343492777337204179724, 6.61497057113220838361920165158, 8.071416084937120999405606330586, 8.680265915687738043015894417363, 9.430298773506168656487379834590

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