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Properties

Label 2-820-820.723-c0-0-0
Degree $2$
Conductor $820$
Sign $-0.296 + 0.954i$
Analytic cond. $0.409233$
Root an. cond. $0.639713$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.891 − 0.453i)5-s + (−0.309 − 0.951i)8-s + (−0.707 − 0.707i)9-s + (−0.987 + 0.156i)10-s + (0.0366 + 0.152i)13-s + (−0.809 − 0.587i)16-s + (1.47 − 1.26i)17-s + (−0.987 − 0.156i)18-s + (−0.707 + 0.707i)20-s + (0.587 + 0.809i)25-s + (0.119 + 0.101i)26-s + (0.144 + 1.84i)29-s − 32-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (0.309 − 0.951i)4-s + (−0.891 − 0.453i)5-s + (−0.309 − 0.951i)8-s + (−0.707 − 0.707i)9-s + (−0.987 + 0.156i)10-s + (0.0366 + 0.152i)13-s + (−0.809 − 0.587i)16-s + (1.47 − 1.26i)17-s + (−0.987 − 0.156i)18-s + (−0.707 + 0.707i)20-s + (0.587 + 0.809i)25-s + (0.119 + 0.101i)26-s + (0.144 + 1.84i)29-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.296 + 0.954i$
Analytic conductor: \(0.409233\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (723, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :0),\ -0.296 + 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.243414249\)
\(L(\frac12)\) \(\approx\) \(1.243414249\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (0.891 + 0.453i)T \)
41 \( 1 + (0.587 + 0.809i)T \)
good3 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.453 - 0.891i)T^{2} \)
11 \( 1 + (0.156 - 0.987i)T^{2} \)
13 \( 1 + (-0.0366 - 0.152i)T + (-0.891 + 0.453i)T^{2} \)
17 \( 1 + (-1.47 + 1.26i)T + (0.156 - 0.987i)T^{2} \)
19 \( 1 + (0.453 - 0.891i)T^{2} \)
23 \( 1 + (0.951 + 0.309i)T^{2} \)
29 \( 1 + (-0.144 - 1.84i)T + (-0.987 + 0.156i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.863 - 1.69i)T + (-0.587 + 0.809i)T^{2} \)
43 \( 1 + (0.309 - 0.951i)T^{2} \)
47 \( 1 + (-0.453 - 0.891i)T^{2} \)
53 \( 1 + (-1.10 + 1.29i)T + (-0.156 - 0.987i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.142 - 0.896i)T + (-0.951 + 0.309i)T^{2} \)
67 \( 1 + (-0.987 + 0.156i)T^{2} \)
71 \( 1 + (-0.156 + 0.987i)T^{2} \)
73 \( 1 + 1.78iT - T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.744 + 0.178i)T + (0.891 + 0.453i)T^{2} \)
97 \( 1 + (-0.101 - 1.29i)T + (-0.987 + 0.156i)T^{2} \)
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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

−10.33267957828280679011896831285, −9.422832548194839667904591338831, −8.648262399809128188115535507473, −7.50698638207942857609908215016, −6.61295097228385791095331541712, −5.45628048238115862328868697535, −4.80205476243289120223052158975, −3.57478140022757017683870736987, −2.98081962808169332441617228833, −1.07626298196868438396794698668, 2.47104627426295601487773211216, 3.52101091552526757700306416923, 4.32412529742208135276792655770, 5.53205435512992920096143688921, 6.16981173853416644749605110002, 7.36155495620767137974834591441, 7.989481851120618822861658529915, 8.489463522421318792018861345443, 9.989663795340947653298106401366, 10.95466916580584459212862502119

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