Properties

Label 2-1045-1.1-c5-0-127
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  − 1.49·2-s + 1.90·3-s − 29.7·4-s + 25·5-s − 2.84·6-s + 61.7·7-s + 92.3·8-s − 239.·9-s − 37.3·10-s + 121·11-s − 56.6·12-s + 947.·13-s − 92.2·14-s + 47.6·15-s + 814.·16-s + 1.44e3·17-s + 357.·18-s + 361·19-s − 744.·20-s + 117.·21-s − 180.·22-s + 1.20e3·23-s + 175.·24-s + 625·25-s − 1.41e3·26-s − 918.·27-s − 1.83e3·28-s + ⋯
L(s)  = 1  − 0.264·2-s + 0.122·3-s − 0.930·4-s + 0.447·5-s − 0.0322·6-s + 0.476·7-s + 0.509·8-s − 0.985·9-s − 0.118·10-s + 0.301·11-s − 0.113·12-s + 1.55·13-s − 0.125·14-s + 0.0546·15-s + 0.795·16-s + 1.20·17-s + 0.260·18-s + 0.229·19-s − 0.416·20-s + 0.0581·21-s − 0.0796·22-s + 0.475·23-s + 0.0622·24-s + 0.200·25-s − 0.410·26-s − 0.242·27-s − 0.442·28-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

\(L(3)\) \(\approx\) \(2.213316884\)
\(L(\frac12)\) \(\approx\) \(2.213316884\)
\(L(\frac{7}{2})\) 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 \( 1 + 1.49T + 32T^{2} \)
3 \( 1 - 1.90T + 243T^{2} \)
7 \( 1 - 61.7T + 1.68e4T^{2} \)
13 \( 1 - 947.T + 3.71e5T^{2} \)
17 \( 1 - 1.44e3T + 1.41e6T^{2} \)
23 \( 1 - 1.20e3T + 6.43e6T^{2} \)
29 \( 1 - 6.33e3T + 2.05e7T^{2} \)
31 \( 1 + 4.62e3T + 2.86e7T^{2} \)
37 \( 1 - 1.61e4T + 6.93e7T^{2} \)
41 \( 1 + 1.33e4T + 1.15e8T^{2} \)
43 \( 1 + 1.22e3T + 1.47e8T^{2} \)
47 \( 1 - 2.44e3T + 2.29e8T^{2} \)
53 \( 1 + 3.47e4T + 4.18e8T^{2} \)
59 \( 1 - 2.41e4T + 7.14e8T^{2} \)
61 \( 1 - 3.47e4T + 8.44e8T^{2} \)
67 \( 1 + 2.36e4T + 1.35e9T^{2} \)
71 \( 1 - 1.75e4T + 1.80e9T^{2} \)
73 \( 1 - 4.18e4T + 2.07e9T^{2} \)
79 \( 1 + 1.23e4T + 3.07e9T^{2} \)
83 \( 1 - 8.94e4T + 3.93e9T^{2} \)
89 \( 1 - 5.33e4T + 5.58e9T^{2} \)
97 \( 1 + 1.50e5T + 8.58e9T^{2} \)
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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

−9.168094980937516523441916118586, −8.300582893352280600674015967130, −8.014091132810444541539661656701, −6.54596756674893426180408565771, −5.67895040131506204056411375259, −4.99322865484960868333659143781, −3.83590897611522699099231050469, −3.00850877032556388585142687713, −1.48047794973943003792855985905, −0.74741344583342832842616458359, 0.74741344583342832842616458359, 1.48047794973943003792855985905, 3.00850877032556388585142687713, 3.83590897611522699099231050469, 4.99322865484960868333659143781, 5.67895040131506204056411375259, 6.54596756674893426180408565771, 8.014091132810444541539661656701, 8.300582893352280600674015967130, 9.168094980937516523441916118586

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