Properties

Label 2-1045-1.1-c1-0-13
Degree $2$
Conductor $1045$
Sign $1$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
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  + 0.759·2-s − 2.25·3-s − 1.42·4-s + 5-s − 1.71·6-s + 4.95·7-s − 2.59·8-s + 2.09·9-s + 0.759·10-s − 11-s + 3.21·12-s + 0.499·13-s + 3.75·14-s − 2.25·15-s + 0.874·16-s − 4.34·17-s + 1.59·18-s − 19-s − 1.42·20-s − 11.1·21-s − 0.759·22-s − 2.78·23-s + 5.86·24-s + 25-s + 0.378·26-s + 2.03·27-s − 7.05·28-s + ⋯
L(s)  = 1  + 0.536·2-s − 1.30·3-s − 0.711·4-s + 0.447·5-s − 0.699·6-s + 1.87·7-s − 0.918·8-s + 0.699·9-s + 0.240·10-s − 0.301·11-s + 0.928·12-s + 0.138·13-s + 1.00·14-s − 0.583·15-s + 0.218·16-s − 1.05·17-s + 0.375·18-s − 0.229·19-s − 0.318·20-s − 2.44·21-s − 0.161·22-s − 0.581·23-s + 1.19·24-s + 0.200·25-s + 0.0743·26-s + 0.391·27-s − 1.33·28-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.322726495\)
\(L(\frac12)\) \(\approx\) \(1.322726495\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.759T + 2T^{2} \)
3 \( 1 + 2.25T + 3T^{2} \)
7 \( 1 - 4.95T + 7T^{2} \)
13 \( 1 - 0.499T + 13T^{2} \)
17 \( 1 + 4.34T + 17T^{2} \)
23 \( 1 + 2.78T + 23T^{2} \)
29 \( 1 - 8.84T + 29T^{2} \)
31 \( 1 - 1.67T + 31T^{2} \)
37 \( 1 + 1.81T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 - 2.65T + 43T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 - 2.84T + 67T^{2} \)
71 \( 1 + 13.3T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + 7.08T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 + 7.25T + 89T^{2} \)
97 \( 1 - 0.0194T + 97T^{2} \)
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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

−10.30728938952265820501309085368, −8.918802811314968053009792563308, −8.417989894292391253605665046994, −7.28764444058576914782079793463, −6.04670338689007252659706132710, −5.59621799603212288411435505408, −4.60696061634001472260848610240, −4.39533004064344175898891492634, −2.42239534740499062902560426424, −0.909463286729259167564313350100, 0.909463286729259167564313350100, 2.42239534740499062902560426424, 4.39533004064344175898891492634, 4.60696061634001472260848610240, 5.59621799603212288411435505408, 6.04670338689007252659706132710, 7.28764444058576914782079793463, 8.417989894292391253605665046994, 8.918802811314968053009792563308, 10.30728938952265820501309085368

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