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Properties

Label 2-820-820.483-c1-0-65
Degree $2$
Conductor $820$
Sign $0.451 + 0.892i$
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  + (−1.35 + 0.401i)2-s + 2.73i·3-s + (1.67 − 1.08i)4-s + (−2.07 + 0.834i)5-s + (−1.10 − 3.71i)6-s + 0.506·7-s + (−1.83 + 2.15i)8-s − 4.50·9-s + (2.47 − 1.96i)10-s + (−0.975 − 0.975i)11-s + (2.98 + 4.59i)12-s − 0.970·13-s + (−0.686 + 0.203i)14-s + (−2.28 − 5.68i)15-s + (1.62 − 3.65i)16-s − 2.65·17-s + ⋯
L(s)  = 1  + (−0.958 + 0.284i)2-s + 1.58i·3-s + (0.838 − 0.544i)4-s + (−0.927 + 0.373i)5-s + (−0.449 − 1.51i)6-s + 0.191·7-s + (−0.649 + 0.760i)8-s − 1.50·9-s + (0.783 − 0.621i)10-s + (−0.294 − 0.294i)11-s + (0.861 + 1.32i)12-s − 0.269·13-s + (−0.183 + 0.0543i)14-s + (−0.590 − 1.46i)15-s + (0.406 − 0.913i)16-s − 0.644·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.451 + 0.892i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (483, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.451 + 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0599865 - 0.0368795i\)
\(L(\frac12)\) \(\approx\) \(0.0599865 - 0.0368795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 - 0.401i)T \)
5 \( 1 + (2.07 - 0.834i)T \)
41 \( 1 + (5.89 - 2.49i)T \)
good3 \( 1 - 2.73iT - 3T^{2} \)
7 \( 1 - 0.506T + 7T^{2} \)
11 \( 1 + (0.975 + 0.975i)T + 11iT^{2} \)
13 \( 1 + 0.970T + 13T^{2} \)
17 \( 1 + 2.65T + 17T^{2} \)
19 \( 1 + (0.779 + 0.779i)T + 19iT^{2} \)
23 \( 1 + (0.0275 - 0.0275i)T - 23iT^{2} \)
29 \( 1 + (5.29 + 5.29i)T + 29iT^{2} \)
31 \( 1 + 5.80iT - 31T^{2} \)
37 \( 1 + (0.173 - 0.173i)T - 37iT^{2} \)
43 \( 1 + (-2.73 - 2.73i)T + 43iT^{2} \)
47 \( 1 + 2.79iT - 47T^{2} \)
53 \( 1 - 1.35T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 - 7.27iT - 61T^{2} \)
67 \( 1 - 3.00iT - 67T^{2} \)
71 \( 1 + (0.285 + 0.285i)T + 71iT^{2} \)
73 \( 1 + (-9.38 + 9.38i)T - 73iT^{2} \)
79 \( 1 + (-5.35 + 5.35i)T - 79iT^{2} \)
83 \( 1 + (2.48 - 2.48i)T - 83iT^{2} \)
89 \( 1 + (-8.00 - 8.00i)T + 89iT^{2} \)
97 \( 1 + 5.38T + 97T^{2} \)
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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

−9.991698938452920627287303248252, −9.356393129947178723490378976397, −8.465270891040281716467404727865, −7.81617367768487653021512358588, −6.76139302393208269181460330081, −5.64265374614732062711005266529, −4.63483723845441655207426186121, −3.68284666798405707972400631888, −2.52228658487763574044197161308, −0.04892491152279590931348767519, 1.31936139821019626093579791029, 2.35195611884676818782108313095, 3.63743064636728921617867993823, 5.19586075755976441201380863579, 6.62381157549199339939186838004, 7.12802166057568206420392100156, 7.889334289463982751482133704142, 8.452174530874722123175298287764, 9.254533672353784350695727367539, 10.54808092341992079929804626428

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