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Properties

Label 2-820-41.23-c1-0-12
Degree $2$
Conductor $820$
Sign $-0.806 + 0.590i$
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  − 3.16i·3-s + (0.809 − 0.587i)5-s + (2.82 + 0.919i)7-s − 7.01·9-s + (1.28 − 1.77i)11-s + (−2.68 + 0.873i)13-s + (−1.86 − 2.56i)15-s + (4.70 − 6.47i)17-s + (−3.09 − 1.00i)19-s + (2.90 − 8.95i)21-s + (−0.383 − 1.18i)23-s + (0.309 − 0.951i)25-s + 12.7i·27-s + (−4.32 − 5.95i)29-s + (0.245 + 0.178i)31-s + ⋯
L(s)  = 1  − 1.82i·3-s + (0.361 − 0.262i)5-s + (1.06 + 0.347i)7-s − 2.33·9-s + (0.388 − 0.535i)11-s + (−0.745 + 0.242i)13-s + (−0.480 − 0.661i)15-s + (1.14 − 1.57i)17-s + (−0.708 − 0.230i)19-s + (0.634 − 1.95i)21-s + (−0.0800 − 0.246i)23-s + (0.0618 − 0.190i)25-s + 2.44i·27-s + (−0.803 − 1.10i)29-s + (0.0441 + 0.0320i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.806 + 0.590i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.806 + 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.508526 - 1.55535i\)
\(L(\frac12)\) \(\approx\) \(0.508526 - 1.55535i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.77 - 5.77i)T \)
good3 \( 1 + 3.16iT - 3T^{2} \)
7 \( 1 + (-2.82 - 0.919i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.28 + 1.77i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.68 - 0.873i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.70 + 6.47i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.09 + 1.00i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.383 + 1.18i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.32 + 5.95i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.245 - 0.178i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.66 - 1.20i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-2.95 - 9.10i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.83 - 0.922i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.26 - 5.87i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.86 - 11.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-4.37 + 13.4i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.52 + 4.85i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-5.09 + 7.01i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + 3.50T + 73T^{2} \)
79 \( 1 - 2.13iT - 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + (-16.8 - 5.47i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.51 - 6.21i)T + (-29.9 + 92.2i)T^{2} \)
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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

−9.688053128274674281214679445094, −8.826580781990899495635765198020, −7.946073397302038869963054884264, −7.48462675366663204074940425951, −6.45796815931192617082947939207, −5.67307867846315158882817788736, −4.75319433084003226281241200577, −2.83494531978475634677524866946, −1.94716210270568699325560191059, −0.829835291020934456255201148991, 1.97685476110953538343583501567, 3.53978617824478387774324450520, 4.18397487232994246767561480765, 5.18079791892217939160022582307, 5.76791542593866852199486177422, 7.23510687807574375318107263057, 8.290872244498485236358663327053, 9.002123639486996959700209757774, 10.05390617952302468642858747855, 10.34298656611924702690509117105

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