Properties

Label 2-1045-1.1-c3-0-151
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $61.6569$
Root an. cond. $7.85219$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.82·2-s + 5.45·3-s − 4.67·4-s + 5·5-s − 9.94·6-s + 19.7·7-s + 23.1·8-s + 2.78·9-s − 9.11·10-s + 11·11-s − 25.5·12-s − 35.9·13-s − 35.9·14-s + 27.2·15-s − 4.69·16-s − 72.8·17-s − 5.08·18-s + 19·19-s − 23.3·20-s + 107.·21-s − 20.0·22-s − 204.·23-s + 126.·24-s + 25·25-s + 65.4·26-s − 132.·27-s − 92.1·28-s + ⋯
L(s)  = 1  − 0.644·2-s + 1.05·3-s − 0.584·4-s + 0.447·5-s − 0.676·6-s + 1.06·7-s + 1.02·8-s + 0.103·9-s − 0.288·10-s + 0.301·11-s − 0.614·12-s − 0.766·13-s − 0.685·14-s + 0.469·15-s − 0.0733·16-s − 1.03·17-s − 0.0665·18-s + 0.229·19-s − 0.261·20-s + 1.11·21-s − 0.194·22-s − 1.85·23-s + 1.07·24-s + 0.200·25-s + 0.493·26-s − 0.941·27-s − 0.622·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(61.6569\)
Root analytic conductor: \(7.85219\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1045,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 5T \)
11 \( 1 - 11T \)
19 \( 1 - 19T \)
good2 \( 1 + 1.82T + 8T^{2} \)
3 \( 1 - 5.45T + 27T^{2} \)
7 \( 1 - 19.7T + 343T^{2} \)
13 \( 1 + 35.9T + 2.19e3T^{2} \)
17 \( 1 + 72.8T + 4.91e3T^{2} \)
23 \( 1 + 204.T + 1.21e4T^{2} \)
29 \( 1 - 206.T + 2.43e4T^{2} \)
31 \( 1 + 292.T + 2.97e4T^{2} \)
37 \( 1 + 79.3T + 5.06e4T^{2} \)
41 \( 1 + 85.3T + 6.89e4T^{2} \)
43 \( 1 + 276.T + 7.95e4T^{2} \)
47 \( 1 - 20.1T + 1.03e5T^{2} \)
53 \( 1 - 514.T + 1.48e5T^{2} \)
59 \( 1 + 613.T + 2.05e5T^{2} \)
61 \( 1 - 487.T + 2.26e5T^{2} \)
67 \( 1 - 578.T + 3.00e5T^{2} \)
71 \( 1 + 637.T + 3.57e5T^{2} \)
73 \( 1 - 582.T + 3.89e5T^{2} \)
79 \( 1 + 129.T + 4.93e5T^{2} \)
83 \( 1 + 860.T + 5.71e5T^{2} \)
89 \( 1 - 845.T + 7.04e5T^{2} \)
97 \( 1 - 18.0T + 9.12e5T^{2} \)
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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

−9.015834027336388884731518148863, −8.385863387846987965976708520469, −7.87971132357054710887243853186, −6.92613622278000801190416652794, −5.54093754959187883181279320568, −4.64587785071477958870069786917, −3.76850257284872217987627438100, −2.32120647854273339213290598690, −1.61636906542104148984014410611, 0, 1.61636906542104148984014410611, 2.32120647854273339213290598690, 3.76850257284872217987627438100, 4.64587785071477958870069786917, 5.54093754959187883181279320568, 6.92613622278000801190416652794, 7.87971132357054710887243853186, 8.385863387846987965976708520469, 9.015834027336388884731518148863

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