Properties

Label 2-1045-1045.379-c0-0-0
Degree $2$
Conductor $1045$
Sign $0.835 - 0.550i$
Analytic cond. $0.521522$
Root an. cond. $0.722165$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0966 + 0.297i)2-s + (−1.44 − 1.04i)3-s + (0.729 − 0.530i)4-s + (−0.309 + 0.951i)5-s + (0.172 − 0.530i)6-s + (0.481 + 0.349i)8-s + (0.672 + 2.06i)9-s − 0.312·10-s + (−0.951 + 0.309i)11-s − 1.60·12-s + (0.610 + 1.87i)13-s + (1.44 − 1.04i)15-s + (0.221 − 0.680i)16-s + (−0.550 + 0.400i)18-s + (0.809 + 0.587i)19-s + (0.278 + 0.857i)20-s + ⋯
L(s)  = 1  + (0.0966 + 0.297i)2-s + (−1.44 − 1.04i)3-s + (0.729 − 0.530i)4-s + (−0.309 + 0.951i)5-s + (0.172 − 0.530i)6-s + (0.481 + 0.349i)8-s + (0.672 + 2.06i)9-s − 0.312·10-s + (−0.951 + 0.309i)11-s − 1.60·12-s + (0.610 + 1.87i)13-s + (1.44 − 1.04i)15-s + (0.221 − 0.680i)16-s + (−0.550 + 0.400i)18-s + (0.809 + 0.587i)19-s + (0.278 + 0.857i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.835 - 0.550i$
Analytic conductor: \(0.521522\)
Root analytic conductor: \(0.722165\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :0),\ 0.835 - 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7375205957\)
\(L(\frac12)\) \(\approx\) \(0.7375205957\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good2 \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \)
3 \( 1 + (1.44 + 1.04i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.610 - 1.87i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.734 + 0.533i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.437 - 1.34i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - 0.907T + T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.0966 + 0.297i)T + (-0.809 + 0.587i)T^{2} \)
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

−10.65542476454483898268835327365, −9.673299755069104978606101227983, −8.034002820146800907149286370065, −7.19651142471171418179539945298, −6.90730714384291636260161384033, −6.04978187364051797597068320119, −5.49617670156035972805251495085, −4.29623414056635024986549847591, −2.49537051034919396804455674667, −1.51047484091680220904798594506, 0.850581854587085936778647363934, 3.01678485971927833662483759855, 3.88248815049785075782786717084, 5.00917109128894673950886024309, 5.48104639492449301787903165828, 6.38189278490990326118742080308, 7.66720376459383121030968188264, 8.293036846541710386981559902991, 9.532760976411732242530638449783, 10.32475944482042833180746257003

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