Properties

Label 2-1045-1.1-c3-0-84
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  − 2.35·2-s − 6.45·3-s − 2.44·4-s − 5·5-s + 15.2·6-s + 27.1·7-s + 24.6·8-s + 14.6·9-s + 11.7·10-s − 11·11-s + 15.7·12-s − 62.2·13-s − 63.9·14-s + 32.2·15-s − 38.4·16-s − 24.6·17-s − 34.4·18-s + 19·19-s + 12.2·20-s − 175.·21-s + 25.9·22-s − 87.4·23-s − 158.·24-s + 25·25-s + 146.·26-s + 79.8·27-s − 66.4·28-s + ⋯
L(s)  = 1  − 0.833·2-s − 1.24·3-s − 0.305·4-s − 0.447·5-s + 1.03·6-s + 1.46·7-s + 1.08·8-s + 0.541·9-s + 0.372·10-s − 0.301·11-s + 0.379·12-s − 1.32·13-s − 1.22·14-s + 0.555·15-s − 0.600·16-s − 0.351·17-s − 0.451·18-s + 0.229·19-s + 0.136·20-s − 1.82·21-s + 0.251·22-s − 0.792·23-s − 1.35·24-s + 0.200·25-s + 1.10·26-s + 0.568·27-s − 0.448·28-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: $\chi_{1045} (1, \cdot )$
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 \( 1 + 2.35T + 8T^{2} \)
3 \( 1 + 6.45T + 27T^{2} \)
7 \( 1 - 27.1T + 343T^{2} \)
13 \( 1 + 62.2T + 2.19e3T^{2} \)
17 \( 1 + 24.6T + 4.91e3T^{2} \)
23 \( 1 + 87.4T + 1.21e4T^{2} \)
29 \( 1 - 221.T + 2.43e4T^{2} \)
31 \( 1 + 214.T + 2.97e4T^{2} \)
37 \( 1 + 192.T + 5.06e4T^{2} \)
41 \( 1 - 323.T + 6.89e4T^{2} \)
43 \( 1 - 382.T + 7.95e4T^{2} \)
47 \( 1 + 12.8T + 1.03e5T^{2} \)
53 \( 1 + 345.T + 1.48e5T^{2} \)
59 \( 1 - 318.T + 2.05e5T^{2} \)
61 \( 1 + 461.T + 2.26e5T^{2} \)
67 \( 1 - 868.T + 3.00e5T^{2} \)
71 \( 1 - 617.T + 3.57e5T^{2} \)
73 \( 1 - 1.12e3T + 3.89e5T^{2} \)
79 \( 1 - 611.T + 4.93e5T^{2} \)
83 \( 1 - 302.T + 5.71e5T^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 + 599.T + 9.12e5T^{2} \)
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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

−9.145770955261730056603446277098, −8.145604059315839449768701060723, −7.69706173836785590688187809809, −6.78313121337936440611725836200, −5.40364578579137579320729303672, −4.92353624222568201762753969782, −4.17718707932499737501949141034, −2.21745890904511667242147110343, −0.947996524377135255923814565989, 0, 0.947996524377135255923814565989, 2.21745890904511667242147110343, 4.17718707932499737501949141034, 4.92353624222568201762753969782, 5.40364578579137579320729303672, 6.78313121337936440611725836200, 7.69706173836785590688187809809, 8.145604059315839449768701060723, 9.145770955261730056603446277098

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