Properties

Label 2-1045-1.1-c5-0-224
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $167.601$
Root an. cond. $12.9460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.7·2-s − 1.89·3-s + 84.4·4-s + 25·5-s + 20.4·6-s + 234.·7-s − 566.·8-s − 239.·9-s − 269.·10-s + 121·11-s − 160.·12-s + 279.·13-s − 2.53e3·14-s − 47.4·15-s + 3.40e3·16-s − 1.37e3·17-s + 2.58e3·18-s − 361·19-s + 2.11e3·20-s − 444.·21-s − 1.30e3·22-s − 1.80e3·23-s + 1.07e3·24-s + 625·25-s − 3.01e3·26-s + 914.·27-s + 1.98e4·28-s + ⋯
L(s)  = 1  − 1.90·2-s − 0.121·3-s + 2.63·4-s + 0.447·5-s + 0.232·6-s + 1.80·7-s − 3.12·8-s − 0.985·9-s − 0.853·10-s + 0.301·11-s − 0.321·12-s + 0.458·13-s − 3.45·14-s − 0.0544·15-s + 3.32·16-s − 1.15·17-s + 1.87·18-s − 0.229·19-s + 1.18·20-s − 0.220·21-s − 0.575·22-s − 0.710·23-s + 0.380·24-s + 0.200·25-s − 0.874·26-s + 0.241·27-s + 4.77·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(167.601\)
Root analytic conductor: \(12.9460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 25T \)
11 \( 1 - 121T \)
19 \( 1 + 361T \)
good2 \( 1 + 10.7T + 32T^{2} \)
3 \( 1 + 1.89T + 243T^{2} \)
7 \( 1 - 234.T + 1.68e4T^{2} \)
13 \( 1 - 279.T + 3.71e5T^{2} \)
17 \( 1 + 1.37e3T + 1.41e6T^{2} \)
23 \( 1 + 1.80e3T + 6.43e6T^{2} \)
29 \( 1 - 3.12e3T + 2.05e7T^{2} \)
31 \( 1 + 2.00e3T + 2.86e7T^{2} \)
37 \( 1 - 1.08e4T + 6.93e7T^{2} \)
41 \( 1 + 8.05e3T + 1.15e8T^{2} \)
43 \( 1 - 8.59e3T + 1.47e8T^{2} \)
47 \( 1 - 9.53e3T + 2.29e8T^{2} \)
53 \( 1 + 8.64e3T + 4.18e8T^{2} \)
59 \( 1 + 4.62e4T + 7.14e8T^{2} \)
61 \( 1 + 2.46e3T + 8.44e8T^{2} \)
67 \( 1 + 6.65e4T + 1.35e9T^{2} \)
71 \( 1 + 1.49e4T + 1.80e9T^{2} \)
73 \( 1 + 6.62e4T + 2.07e9T^{2} \)
79 \( 1 - 1.23e3T + 3.07e9T^{2} \)
83 \( 1 - 3.16e4T + 3.93e9T^{2} \)
89 \( 1 - 6.21e3T + 5.58e9T^{2} \)
97 \( 1 - 6.04e3T + 8.58e9T^{2} \)
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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

−8.773932254736321610141661648672, −8.179949206406029175762173379837, −7.51614595002082953205460933387, −6.40678308073990105825284403945, −5.75661214051359158883156328042, −4.47712593250652163259696884739, −2.72935146859863878094447970451, −1.90312035513134563007788335648, −1.17054760462917378336864077426, 0, 1.17054760462917378336864077426, 1.90312035513134563007788335648, 2.72935146859863878094447970451, 4.47712593250652163259696884739, 5.75661214051359158883156328042, 6.40678308073990105825284403945, 7.51614595002082953205460933387, 8.179949206406029175762173379837, 8.773932254736321610141661648672

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