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Properties

Label 2-820-820.43-c1-0-63
Degree $2$
Conductor $820$
Sign $0.925 + 0.379i$
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  + (0.226 + 1.39i)2-s − 1.94i·3-s + (−1.89 + 0.632i)4-s + (−2.19 + 0.409i)5-s + (2.71 − 0.440i)6-s + (−1.22 + 3.77i)7-s + (−1.31 − 2.50i)8-s − 0.773·9-s + (−1.06 − 2.97i)10-s + (−0.621 − 3.92i)11-s + (1.22 + 3.68i)12-s + (−0.213 − 0.656i)13-s + (−5.54 − 0.857i)14-s + (0.795 + 4.27i)15-s + (3.19 − 2.40i)16-s + (2.52 + 1.83i)17-s + ⋯
L(s)  = 1  + (0.160 + 0.987i)2-s − 1.12i·3-s + (−0.948 + 0.316i)4-s + (−0.983 + 0.183i)5-s + (1.10 − 0.179i)6-s + (−0.463 + 1.42i)7-s + (−0.464 − 0.885i)8-s − 0.257·9-s + (−0.338 − 0.941i)10-s + (−0.187 − 1.18i)11-s + (0.354 + 1.06i)12-s + (−0.0591 − 0.182i)13-s + (−1.48 − 0.229i)14-s + (0.205 + 1.10i)15-s + (0.799 − 0.600i)16-s + (0.612 + 0.445i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 + 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.925 + 0.379i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.925 + 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02117 - 0.201505i\)
\(L(\frac12)\) \(\approx\) \(1.02117 - 0.201505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.226 - 1.39i)T \)
5 \( 1 + (2.19 - 0.409i)T \)
41 \( 1 + (6.12 - 1.88i)T \)
good3 \( 1 + 1.94iT - 3T^{2} \)
7 \( 1 + (1.22 - 3.77i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.621 + 3.92i)T + (-10.4 + 3.39i)T^{2} \)
13 \( 1 + (0.213 + 0.656i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.52 - 1.83i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-7.31 + 3.72i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (-0.124 - 0.0634i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (7.09 + 1.12i)T + (27.5 + 8.96i)T^{2} \)
31 \( 1 + (-5.70 + 7.85i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.340 + 2.15i)T + (-35.1 - 11.4i)T^{2} \)
43 \( 1 + (1.39 - 2.72i)T + (-25.2 - 34.7i)T^{2} \)
47 \( 1 + (-5.29 + 1.71i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.05 + 5.12i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.88 + 11.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.25 - 1.05i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-0.542 - 0.746i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-2.55 - 16.1i)T + (-67.5 + 21.9i)T^{2} \)
73 \( 1 + (0.650 - 0.650i)T - 73iT^{2} \)
79 \( 1 + (-4.91 + 4.91i)T - 79iT^{2} \)
83 \( 1 + (-2.84 + 2.84i)T - 83iT^{2} \)
89 \( 1 + (-5.28 + 2.69i)T + (52.3 - 72.0i)T^{2} \)
97 \( 1 + (-0.182 + 0.132i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.873182544369685101479072497530, −8.963807684635940789107631475471, −8.141192842670188386121501507151, −7.66170749696726441792866006878, −6.75645310863503748118097154487, −5.92763852831606573068759165198, −5.24660438861152455148800088018, −3.68467731967406724846207946465, −2.76403778974160991289175616832, −0.60497128920846285695610656509, 1.17450734416158947837350698189, 3.26979323741602351345896432882, 3.78039116961513601913314135304, 4.59040959517907043195273126767, 5.25331972309596922809537298910, 7.13170648974160073817821411251, 7.69675946781578917117913544850, 9.022060880117530158214831541935, 9.819948634228058543127440886076, 10.24609230977098442998583678628

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