Properties

Label 2-1045-1.1-c3-0-78
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  + 1.78·2-s − 7.45·3-s − 4.80·4-s − 5·5-s − 13.3·6-s − 20.9·7-s − 22.8·8-s + 28.5·9-s − 8.94·10-s − 11·11-s + 35.7·12-s + 37.7·13-s − 37.4·14-s + 37.2·15-s − 2.52·16-s + 119.·17-s + 51.0·18-s + 19·19-s + 24.0·20-s + 156.·21-s − 19.6·22-s − 24.1·23-s + 170.·24-s + 25·25-s + 67.5·26-s − 11.4·27-s + 100.·28-s + ⋯
L(s)  = 1  + 0.632·2-s − 1.43·3-s − 0.600·4-s − 0.447·5-s − 0.906·6-s − 1.13·7-s − 1.01·8-s + 1.05·9-s − 0.282·10-s − 0.301·11-s + 0.860·12-s + 0.805·13-s − 0.714·14-s + 0.641·15-s − 0.0395·16-s + 1.70·17-s + 0.668·18-s + 0.229·19-s + 0.268·20-s + 1.62·21-s − 0.190·22-s − 0.218·23-s + 1.45·24-s + 0.200·25-s + 0.509·26-s − 0.0812·27-s + 0.678·28-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 \( 1 - 1.78T + 8T^{2} \)
3 \( 1 + 7.45T + 27T^{2} \)
7 \( 1 + 20.9T + 343T^{2} \)
13 \( 1 - 37.7T + 2.19e3T^{2} \)
17 \( 1 - 119.T + 4.91e3T^{2} \)
23 \( 1 + 24.1T + 1.21e4T^{2} \)
29 \( 1 - 1.45T + 2.43e4T^{2} \)
31 \( 1 + 7.62T + 2.97e4T^{2} \)
37 \( 1 + 402.T + 5.06e4T^{2} \)
41 \( 1 - 278.T + 6.89e4T^{2} \)
43 \( 1 + 228.T + 7.95e4T^{2} \)
47 \( 1 - 409.T + 1.03e5T^{2} \)
53 \( 1 - 45.2T + 1.48e5T^{2} \)
59 \( 1 + 203.T + 2.05e5T^{2} \)
61 \( 1 + 196.T + 2.26e5T^{2} \)
67 \( 1 + 81.1T + 3.00e5T^{2} \)
71 \( 1 - 257.T + 3.57e5T^{2} \)
73 \( 1 - 544.T + 3.89e5T^{2} \)
79 \( 1 - 77.8T + 4.93e5T^{2} \)
83 \( 1 + 819.T + 5.71e5T^{2} \)
89 \( 1 - 992.T + 7.04e5T^{2} \)
97 \( 1 - 1.22e3T + 9.12e5T^{2} \)
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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

−9.298798833202570597629045827856, −8.273986532092053583028399036725, −7.16483217874830264901673598126, −6.18698914056639523456468891873, −5.67653592390697738132752269657, −4.94593903772680317484418465629, −3.81624303651395462935782518732, −3.17303429563704023513312296199, −0.932749171450588690272505439689, 0, 0.932749171450588690272505439689, 3.17303429563704023513312296199, 3.81624303651395462935782518732, 4.94593903772680317484418465629, 5.67653592390697738132752269657, 6.18698914056639523456468891873, 7.16483217874830264901673598126, 8.273986532092053583028399036725, 9.298798833202570597629045827856

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