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Properties

Label 2-820-205.178-c1-0-15
Degree $2$
Conductor $820$
Sign $-0.0273 + 0.999i$
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  + (0.585 − 1.41i)3-s + (−1.11 − 1.93i)5-s + (1.32 + 0.549i)7-s + (0.464 + 0.464i)9-s + (5.38 − 2.23i)11-s + (1.03 − 2.48i)13-s + (−3.39 + 0.447i)15-s + (1.27 + 3.07i)17-s + (−5.44 − 2.25i)19-s + (1.55 − 1.55i)21-s + (−0.656 + 0.656i)23-s + (−2.49 + 4.33i)25-s + (5.17 − 2.14i)27-s + (−1.75 + 0.725i)29-s − 4.97i·31-s + ⋯
L(s)  = 1  + (0.338 − 0.816i)3-s + (−0.500 − 0.865i)5-s + (0.501 + 0.207i)7-s + (0.154 + 0.154i)9-s + (1.62 − 0.672i)11-s + (0.285 − 0.690i)13-s + (−0.876 + 0.115i)15-s + (0.308 + 0.745i)17-s + (−1.24 − 0.517i)19-s + (0.339 − 0.339i)21-s + (−0.136 + 0.136i)23-s + (−0.499 + 0.866i)25-s + (0.995 − 0.412i)27-s + (−0.325 + 0.134i)29-s − 0.893i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0273 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0273 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.0273 + 0.999i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.0273 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26539 - 1.30048i\)
\(L(\frac12)\) \(\approx\) \(1.26539 - 1.30048i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 + 1.93i)T \)
41 \( 1 + (6.38 + 0.465i)T \)
good3 \( 1 + (-0.585 + 1.41i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-1.32 - 0.549i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-5.38 + 2.23i)T + (7.77 - 7.77i)T^{2} \)
13 \( 1 + (-1.03 + 2.48i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + (-1.27 - 3.07i)T + (-12.0 + 12.0i)T^{2} \)
19 \( 1 + (5.44 + 2.25i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (0.656 - 0.656i)T - 23iT^{2} \)
29 \( 1 + (1.75 - 0.725i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 4.97iT - 31T^{2} \)
37 \( 1 + (-5.97 + 5.97i)T - 37iT^{2} \)
43 \( 1 - 8.73iT - 43T^{2} \)
47 \( 1 + (5.18 + 12.5i)T + (-33.2 + 33.2i)T^{2} \)
53 \( 1 + (-3.13 - 1.29i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 - 10.4iT - 59T^{2} \)
61 \( 1 + (1.06 + 1.06i)T + 61iT^{2} \)
67 \( 1 + (-0.282 - 0.682i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-5.84 - 14.1i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + 7.69iT - 73T^{2} \)
79 \( 1 + (-1.10 - 2.67i)T + (-55.8 + 55.8i)T^{2} \)
83 \( 1 + (-0.371 - 0.371i)T + 83iT^{2} \)
89 \( 1 + (15.0 - 6.22i)T + (62.9 - 62.9i)T^{2} \)
97 \( 1 + (-3.44 + 1.42i)T + (68.5 - 68.5i)T^{2} \)
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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

−9.914186685281287560065944531711, −8.724126117269184046177176301528, −8.450099336606065000572368230190, −7.62704981752759705264250276366, −6.59774419686194647109861424455, −5.70812581180061112403255733467, −4.47458631990197733694952712208, −3.64504801290225164692230478514, −1.98901890762983911257048872277, −0.979031611774985404506332847889, 1.68803641390250118004557499919, 3.26906694117198740300869022875, 4.10574749917379987548401593101, 4.64784912839743957700998162248, 6.41944640204407119500033979063, 6.84330125205556264726304744112, 7.950365713585246831519484620623, 8.911421632729290248838663520746, 9.646080946963991869511296192830, 10.35308596637015862047987137723

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