Properties

Label 2-1045-1.1-c5-0-71
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  + 8.18·2-s − 3.55·3-s + 34.9·4-s + 25·5-s − 29.1·6-s − 238.·7-s + 23.9·8-s − 230.·9-s + 204.·10-s + 121·11-s − 124.·12-s + 229.·13-s − 1.95e3·14-s − 88.9·15-s − 921.·16-s − 93.8·17-s − 1.88e3·18-s + 361·19-s + 873.·20-s + 849.·21-s + 989.·22-s − 1.10e3·23-s − 85.3·24-s + 625·25-s + 1.87e3·26-s + 1.68e3·27-s − 8.34e3·28-s + ⋯
L(s)  = 1  + 1.44·2-s − 0.228·3-s + 1.09·4-s + 0.447·5-s − 0.330·6-s − 1.84·7-s + 0.132·8-s − 0.947·9-s + 0.646·10-s + 0.301·11-s − 0.249·12-s + 0.376·13-s − 2.66·14-s − 0.102·15-s − 0.899·16-s − 0.0787·17-s − 1.37·18-s + 0.229·19-s + 0.488·20-s + 0.420·21-s + 0.436·22-s − 0.433·23-s − 0.0302·24-s + 0.200·25-s + 0.544·26-s + 0.444·27-s − 2.01·28-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

\(L(3)\) \(\approx\) \(2.767410977\)
\(L(\frac12)\) \(\approx\) \(2.767410977\)
\(L(\frac{7}{2})\) 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 \( 1 - 8.18T + 32T^{2} \)
3 \( 1 + 3.55T + 243T^{2} \)
7 \( 1 + 238.T + 1.68e4T^{2} \)
13 \( 1 - 229.T + 3.71e5T^{2} \)
17 \( 1 + 93.8T + 1.41e6T^{2} \)
23 \( 1 + 1.10e3T + 6.43e6T^{2} \)
29 \( 1 + 849.T + 2.05e7T^{2} \)
31 \( 1 - 4.21e3T + 2.86e7T^{2} \)
37 \( 1 - 1.15e4T + 6.93e7T^{2} \)
41 \( 1 + 1.62e4T + 1.15e8T^{2} \)
43 \( 1 + 5.23e3T + 1.47e8T^{2} \)
47 \( 1 - 1.66e3T + 2.29e8T^{2} \)
53 \( 1 - 2.03e3T + 4.18e8T^{2} \)
59 \( 1 - 1.82e4T + 7.14e8T^{2} \)
61 \( 1 + 9.01e3T + 8.44e8T^{2} \)
67 \( 1 - 1.31e4T + 1.35e9T^{2} \)
71 \( 1 - 3.12e4T + 1.80e9T^{2} \)
73 \( 1 - 5.08e4T + 2.07e9T^{2} \)
79 \( 1 + 4.81e4T + 3.07e9T^{2} \)
83 \( 1 - 9.68e4T + 3.93e9T^{2} \)
89 \( 1 - 2.67e4T + 5.58e9T^{2} \)
97 \( 1 + 3.95e4T + 8.58e9T^{2} \)
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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

−9.343764758029616500490716772360, −8.440712437653149992808541985194, −6.94055862424762558928480315620, −6.25262978330978084225013622699, −5.90770733171622558217541717967, −4.94939683115632869890664177818, −3.77070767889313575761934446648, −3.16433550199670104851050463875, −2.34622232204145269692126973428, −0.54776394743374383083129262449, 0.54776394743374383083129262449, 2.34622232204145269692126973428, 3.16433550199670104851050463875, 3.77070767889313575761934446648, 4.94939683115632869890664177818, 5.90770733171622558217541717967, 6.25262978330978084225013622699, 6.94055862424762558928480315620, 8.440712437653149992808541985194, 9.343764758029616500490716772360

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