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Properties

Label 2-820-41.16-c1-0-2
Degree $2$
Conductor $820$
Sign $-0.663 - 0.748i$
Analytic cond. $6.54773$
Root an. cond. $2.55885$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.753·3-s + (−0.809 − 0.587i)5-s + (0.739 + 2.27i)7-s − 2.43·9-s + (−2.80 + 2.03i)11-s + (−0.656 + 2.02i)13-s + (−0.609 − 0.443i)15-s + (−5.29 + 3.84i)17-s + (−1.41 − 4.35i)19-s + (0.557 + 1.71i)21-s + (−0.448 + 1.38i)23-s + (0.309 + 0.951i)25-s − 4.09·27-s + (4.76 + 3.46i)29-s + (2.02 − 1.46i)31-s + ⋯
L(s)  = 1  + 0.435·3-s + (−0.361 − 0.262i)5-s + (0.279 + 0.860i)7-s − 0.810·9-s + (−0.844 + 0.613i)11-s + (−0.182 + 0.560i)13-s + (−0.157 − 0.114i)15-s + (−1.28 + 0.933i)17-s + (−0.324 − 0.999i)19-s + (0.121 + 0.374i)21-s + (−0.0936 + 0.288i)23-s + (0.0618 + 0.190i)25-s − 0.788·27-s + (0.884 + 0.642i)29-s + (0.363 − 0.263i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.663 - 0.748i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.663 - 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.322574 + 0.717362i\)
\(L(\frac12)\) \(\approx\) \(0.322574 + 0.717362i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (2.68 - 5.81i)T \)
good3 \( 1 - 0.753T + 3T^{2} \)
7 \( 1 + (-0.739 - 2.27i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (2.80 - 2.03i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.656 - 2.02i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (5.29 - 3.84i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.41 + 4.35i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.448 - 1.38i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-4.76 - 3.46i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-2.02 + 1.46i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.84 + 1.34i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (0.282 - 0.869i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (0.667 - 2.05i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.78 - 3.47i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.99 - 6.15i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.69 + 8.28i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-0.970 - 0.704i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (8.53 - 6.20i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 - 9.44T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + (-1.42 - 4.37i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (6.78 + 4.92i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.63454443825520396142565833725, −9.472934618921783572939474863088, −8.670606020393823621197485086905, −8.309843743368505911471160443002, −7.18976880600563380393998423040, −6.18067149720584316067420599527, −5.10218685690311685678035479346, −4.32499442056441450057387646842, −2.86760486540023588878028775351, −2.05321711872182322859261667563, 0.33827189964650575994999571184, 2.39243532294863761572111684261, 3.30619064848506158860470214208, 4.39860181757032987706174782251, 5.44783357257452517297086931784, 6.53696129948416706990182611309, 7.53438429209756373015937992635, 8.213550828531713925080862023444, 8.846178253390250343145194644774, 10.13947804098203363055438222082

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