Properties

Label 2-1045-1.1-c5-0-137
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.78·2-s + 8.91·3-s − 9.10·4-s + 25·5-s + 42.6·6-s + 59.3·7-s − 196.·8-s − 163.·9-s + 119.·10-s + 121·11-s − 81.2·12-s + 370.·13-s + 283.·14-s + 222.·15-s − 649.·16-s + 1.82e3·17-s − 782.·18-s + 361·19-s − 227.·20-s + 529.·21-s + 578.·22-s + 3.17e3·23-s − 1.75e3·24-s + 625·25-s + 1.77e3·26-s − 3.62e3·27-s − 540.·28-s + ⋯
L(s)  = 1  + 0.845·2-s + 0.572·3-s − 0.284·4-s + 0.447·5-s + 0.483·6-s + 0.457·7-s − 1.08·8-s − 0.672·9-s + 0.378·10-s + 0.301·11-s − 0.162·12-s + 0.608·13-s + 0.387·14-s + 0.255·15-s − 0.634·16-s + 1.53·17-s − 0.569·18-s + 0.229·19-s − 0.127·20-s + 0.261·21-s + 0.255·22-s + 1.25·23-s − 0.621·24-s + 0.200·25-s + 0.514·26-s − 0.956·27-s − 0.130·28-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: $\chi_{1045} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.351939405\)
\(L(\frac12)\) \(\approx\) \(4.351939405\)
\(L(\frac{7}{2})\) 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 \( 1 - 4.78T + 32T^{2} \)
3 \( 1 - 8.91T + 243T^{2} \)
7 \( 1 - 59.3T + 1.68e4T^{2} \)
13 \( 1 - 370.T + 3.71e5T^{2} \)
17 \( 1 - 1.82e3T + 1.41e6T^{2} \)
23 \( 1 - 3.17e3T + 6.43e6T^{2} \)
29 \( 1 + 779.T + 2.05e7T^{2} \)
31 \( 1 + 6.60e3T + 2.86e7T^{2} \)
37 \( 1 + 1.23e4T + 6.93e7T^{2} \)
41 \( 1 + 6.37e3T + 1.15e8T^{2} \)
43 \( 1 + 8.44e3T + 1.47e8T^{2} \)
47 \( 1 - 2.01e4T + 2.29e8T^{2} \)
53 \( 1 - 3.97e3T + 4.18e8T^{2} \)
59 \( 1 - 806.T + 7.14e8T^{2} \)
61 \( 1 - 1.44e4T + 8.44e8T^{2} \)
67 \( 1 - 3.38e4T + 1.35e9T^{2} \)
71 \( 1 - 2.29e4T + 1.80e9T^{2} \)
73 \( 1 - 7.06e4T + 2.07e9T^{2} \)
79 \( 1 - 7.60e4T + 3.07e9T^{2} \)
83 \( 1 + 1.63e4T + 3.93e9T^{2} \)
89 \( 1 - 4.75e4T + 5.58e9T^{2} \)
97 \( 1 - 1.05e5T + 8.58e9T^{2} \)
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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

−9.028886970913568653077054726718, −8.556717951054056984863177390330, −7.57550257545897115131333375323, −6.43485554116996117301912287454, −5.43825212663660134195100437600, −5.09823241135014764228677010937, −3.61088414374561506446819243055, −3.30317295640523679154593504217, −2.01007465220685393979710331737, −0.78264104031387687810423151613, 0.78264104031387687810423151613, 2.01007465220685393979710331737, 3.30317295640523679154593504217, 3.61088414374561506446819243055, 5.09823241135014764228677010937, 5.43825212663660134195100437600, 6.43485554116996117301912287454, 7.57550257545897115131333375323, 8.556717951054056984863177390330, 9.028886970913568653077054726718

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