Properties

Label 2-1045-1.1-c3-0-122
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5.49·2-s + 3.98·3-s + 22.1·4-s − 5·5-s − 21.8·6-s + 23.4·7-s − 77.7·8-s − 11.1·9-s + 27.4·10-s − 11·11-s + 88.3·12-s − 8.78·13-s − 128.·14-s − 19.9·15-s + 249.·16-s − 67.5·17-s + 61.0·18-s + 19·19-s − 110.·20-s + 93.5·21-s + 60.4·22-s + 110.·23-s − 309.·24-s + 25·25-s + 48.2·26-s − 151.·27-s + 520.·28-s + ⋯
L(s)  = 1  − 1.94·2-s + 0.766·3-s + 2.77·4-s − 0.447·5-s − 1.48·6-s + 1.26·7-s − 3.43·8-s − 0.411·9-s + 0.868·10-s − 0.301·11-s + 2.12·12-s − 0.187·13-s − 2.46·14-s − 0.342·15-s + 3.90·16-s − 0.963·17-s + 0.799·18-s + 0.229·19-s − 1.23·20-s + 0.972·21-s + 0.585·22-s + 0.999·23-s − 2.63·24-s + 0.200·25-s + 0.364·26-s − 1.08·27-s + 3.51·28-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: $\chi_{1045} (1, \cdot )$
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 \( 1 + 5.49T + 8T^{2} \)
3 \( 1 - 3.98T + 27T^{2} \)
7 \( 1 - 23.4T + 343T^{2} \)
13 \( 1 + 8.78T + 2.19e3T^{2} \)
17 \( 1 + 67.5T + 4.91e3T^{2} \)
23 \( 1 - 110.T + 1.21e4T^{2} \)
29 \( 1 - 203.T + 2.43e4T^{2} \)
31 \( 1 + 239.T + 2.97e4T^{2} \)
37 \( 1 - 315.T + 5.06e4T^{2} \)
41 \( 1 - 320.T + 6.89e4T^{2} \)
43 \( 1 + 181.T + 7.95e4T^{2} \)
47 \( 1 + 473.T + 1.03e5T^{2} \)
53 \( 1 + 21.6T + 1.48e5T^{2} \)
59 \( 1 + 839.T + 2.05e5T^{2} \)
61 \( 1 + 527.T + 2.26e5T^{2} \)
67 \( 1 - 63.2T + 3.00e5T^{2} \)
71 \( 1 + 717.T + 3.57e5T^{2} \)
73 \( 1 - 787.T + 3.89e5T^{2} \)
79 \( 1 - 337.T + 4.93e5T^{2} \)
83 \( 1 + 676.T + 5.71e5T^{2} \)
89 \( 1 + 669.T + 7.04e5T^{2} \)
97 \( 1 - 1.41e3T + 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.986900141281443318108156561024, −8.336613664776332670983564044861, −7.83497700009678223022794654556, −7.19285840627876209388974701195, −6.09523113868482486564227989706, −4.76199709871080112403264244944, −3.13095148977499255323388510372, −2.31616347005034088092043167841, −1.30951308323370640023036175248, 0, 1.30951308323370640023036175248, 2.31616347005034088092043167841, 3.13095148977499255323388510372, 4.76199709871080112403264244944, 6.09523113868482486564227989706, 7.19285840627876209388974701195, 7.83497700009678223022794654556, 8.336613664776332670983564044861, 8.986900141281443318108156561024

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