Properties

Label 2-1045-1.1-c3-0-67
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  − 0.000359·2-s − 7.63·3-s − 7.99·4-s + 5·5-s + 0.00274·6-s − 26.4·7-s + 0.00574·8-s + 31.3·9-s − 0.00179·10-s + 11·11-s + 61.0·12-s − 29.0·13-s + 0.00949·14-s − 38.1·15-s + 63.9·16-s − 65.6·17-s − 0.0112·18-s + 19·19-s − 39.9·20-s + 201.·21-s − 0.00395·22-s − 176.·23-s − 0.0438·24-s + 25·25-s + 0.0104·26-s − 33.0·27-s + 211.·28-s + ⋯
L(s)  = 1  − 0.000126·2-s − 1.46·3-s − 0.999·4-s + 0.447·5-s + 0.000186·6-s − 1.42·7-s + 0.000253·8-s + 1.16·9-s − 5.67e − 5·10-s + 0.301·11-s + 1.46·12-s − 0.619·13-s + 0.000181·14-s − 0.657·15-s + 0.999·16-s − 0.936·17-s − 0.000147·18-s + 0.229·19-s − 0.447·20-s + 2.09·21-s − 3.82e − 5·22-s − 1.60·23-s − 0.000373·24-s + 0.200·25-s + 7.86e−5·26-s − 0.235·27-s + 1.42·28-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 \( 1 + 0.000359T + 8T^{2} \)
3 \( 1 + 7.63T + 27T^{2} \)
7 \( 1 + 26.4T + 343T^{2} \)
13 \( 1 + 29.0T + 2.19e3T^{2} \)
17 \( 1 + 65.6T + 4.91e3T^{2} \)
23 \( 1 + 176.T + 1.21e4T^{2} \)
29 \( 1 - 233.T + 2.43e4T^{2} \)
31 \( 1 - 314.T + 2.97e4T^{2} \)
37 \( 1 + 137.T + 5.06e4T^{2} \)
41 \( 1 - 172.T + 6.89e4T^{2} \)
43 \( 1 - 291.T + 7.95e4T^{2} \)
47 \( 1 - 612.T + 1.03e5T^{2} \)
53 \( 1 + 346.T + 1.48e5T^{2} \)
59 \( 1 + 168.T + 2.05e5T^{2} \)
61 \( 1 - 702.T + 2.26e5T^{2} \)
67 \( 1 + 675.T + 3.00e5T^{2} \)
71 \( 1 + 645.T + 3.57e5T^{2} \)
73 \( 1 - 967.T + 3.89e5T^{2} \)
79 \( 1 + 1.07e3T + 4.93e5T^{2} \)
83 \( 1 + 236.T + 5.71e5T^{2} \)
89 \( 1 + 675.T + 7.04e5T^{2} \)
97 \( 1 + 690.T + 9.12e5T^{2} \)
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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

−9.376771008199200723353920829555, −8.476845998574844195602380543358, −7.14908044292721937995804070181, −6.22371013412712115482965656783, −5.90879796179087015792796250691, −4.77522260730632446897426176327, −4.11471111637507300046865768153, −2.70210729355105462641155818113, −0.860840580904576960409155344707, 0, 0.860840580904576960409155344707, 2.70210729355105462641155818113, 4.11471111637507300046865768153, 4.77522260730632446897426176327, 5.90879796179087015792796250691, 6.22371013412712115482965656783, 7.14908044292721937995804070181, 8.476845998574844195602380543358, 9.376771008199200723353920829555

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