Properties

Label 2-1045-1.1-c3-0-39
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.817·2-s − 5.32·3-s − 7.33·4-s + 5·5-s − 4.35·6-s + 30.6·7-s − 12.5·8-s + 1.40·9-s + 4.08·10-s − 11·11-s + 39.0·12-s − 31.7·13-s + 25.0·14-s − 26.6·15-s + 48.4·16-s + 98.1·17-s + 1.14·18-s + 19·19-s − 36.6·20-s − 163.·21-s − 8.99·22-s − 89.2·23-s + 66.7·24-s + 25·25-s − 25.9·26-s + 136.·27-s − 224.·28-s + ⋯
L(s)  = 1  + 0.288·2-s − 1.02·3-s − 0.916·4-s + 0.447·5-s − 0.296·6-s + 1.65·7-s − 0.553·8-s + 0.0519·9-s + 0.129·10-s − 0.301·11-s + 0.940·12-s − 0.676·13-s + 0.477·14-s − 0.458·15-s + 0.756·16-s + 1.40·17-s + 0.0150·18-s + 0.229·19-s − 0.409·20-s − 1.69·21-s − 0.0871·22-s − 0.808·23-s + 0.567·24-s + 0.200·25-s − 0.195·26-s + 0.972·27-s − 1.51·28-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: \(0\)
Selberg data: \((2,\ 1045,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.400183304\)
\(L(\frac12)\) \(\approx\) \(1.400183304\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.817T + 8T^{2} \)
3 \( 1 + 5.32T + 27T^{2} \)
7 \( 1 - 30.6T + 343T^{2} \)
13 \( 1 + 31.7T + 2.19e3T^{2} \)
17 \( 1 - 98.1T + 4.91e3T^{2} \)
23 \( 1 + 89.2T + 1.21e4T^{2} \)
29 \( 1 + 33.8T + 2.43e4T^{2} \)
31 \( 1 - 64.3T + 2.97e4T^{2} \)
37 \( 1 + 217.T + 5.06e4T^{2} \)
41 \( 1 + 430.T + 6.89e4T^{2} \)
43 \( 1 + 472.T + 7.95e4T^{2} \)
47 \( 1 + 271.T + 1.03e5T^{2} \)
53 \( 1 - 249.T + 1.48e5T^{2} \)
59 \( 1 - 448.T + 2.05e5T^{2} \)
61 \( 1 - 411.T + 2.26e5T^{2} \)
67 \( 1 - 274.T + 3.00e5T^{2} \)
71 \( 1 - 936.T + 3.57e5T^{2} \)
73 \( 1 - 723.T + 3.89e5T^{2} \)
79 \( 1 - 181.T + 4.93e5T^{2} \)
83 \( 1 + 858.T + 5.71e5T^{2} \)
89 \( 1 + 758.T + 7.04e5T^{2} \)
97 \( 1 - 294.T + 9.12e5T^{2} \)
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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

−9.863801789431306708013475503130, −8.450014647733936922031863463301, −8.129005156232646014919400479472, −6.90413414671901514894619720636, −5.63431784473177629071157795183, −5.23787860726851336476139230781, −4.72686312281770334445088044887, −3.42248709202638879089307419410, −1.84463809955383421780114974350, −0.64951090991392475280698687498, 0.64951090991392475280698687498, 1.84463809955383421780114974350, 3.42248709202638879089307419410, 4.72686312281770334445088044887, 5.23787860726851336476139230781, 5.63431784473177629071157795183, 6.90413414671901514894619720636, 8.129005156232646014919400479472, 8.450014647733936922031863463301, 9.863801789431306708013475503130

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