Properties

Label 2-1045-1.1-c1-0-56
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.22·2-s − 1.47·3-s + 2.96·4-s + 5-s − 3.29·6-s − 4.42·7-s + 2.15·8-s − 0.812·9-s + 2.22·10-s − 11-s − 4.38·12-s − 3.92·13-s − 9.86·14-s − 1.47·15-s − 1.13·16-s − 1.61·17-s − 1.81·18-s + 19-s + 2.96·20-s + 6.54·21-s − 2.22·22-s + 0.113·23-s − 3.18·24-s + 25-s − 8.75·26-s + 5.63·27-s − 13.1·28-s + ⋯
L(s)  = 1  + 1.57·2-s − 0.853·3-s + 1.48·4-s + 0.447·5-s − 1.34·6-s − 1.67·7-s + 0.762·8-s − 0.270·9-s + 0.704·10-s − 0.301·11-s − 1.26·12-s − 1.08·13-s − 2.63·14-s − 0.381·15-s − 0.282·16-s − 0.391·17-s − 0.427·18-s + 0.229·19-s + 0.663·20-s + 1.42·21-s − 0.475·22-s + 0.0236·23-s − 0.650·24-s + 0.200·25-s − 1.71·26-s + 1.08·27-s − 2.48·28-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: \(1\)
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 \( 1 - 2.22T + 2T^{2} \)
3 \( 1 + 1.47T + 3T^{2} \)
7 \( 1 + 4.42T + 7T^{2} \)
13 \( 1 + 3.92T + 13T^{2} \)
17 \( 1 + 1.61T + 17T^{2} \)
23 \( 1 - 0.113T + 23T^{2} \)
29 \( 1 - 1.08T + 29T^{2} \)
31 \( 1 + 4.17T + 31T^{2} \)
37 \( 1 + 5.75T + 37T^{2} \)
41 \( 1 - 4.26T + 41T^{2} \)
43 \( 1 - 3.62T + 43T^{2} \)
47 \( 1 + 6.39T + 47T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 - 0.883T + 59T^{2} \)
61 \( 1 - 3.77T + 61T^{2} \)
67 \( 1 + 12.2T + 67T^{2} \)
71 \( 1 - 4.28T + 71T^{2} \)
73 \( 1 + 1.92T + 73T^{2} \)
79 \( 1 + 7.81T + 79T^{2} \)
83 \( 1 + 3.33T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
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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

−9.741063629422071909327386162703, −8.857048145900172957948912625969, −7.16701825764086779906719293890, −6.65186201697669140569987061136, −5.78538816656008188925923333597, −5.37753319588835800476599691347, −4.33510243722719969775879883865, −3.19789646954358869850323480873, −2.47822128914369870164110130036, 0, 2.47822128914369870164110130036, 3.19789646954358869850323480873, 4.33510243722719969775879883865, 5.37753319588835800476599691347, 5.78538816656008188925923333597, 6.65186201697669140569987061136, 7.16701825764086779906719293890, 8.857048145900172957948912625969, 9.741063629422071909327386162703

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