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Properties

Label 2-820-820.207-c1-0-10
Degree $2$
Conductor $820$
Sign $0.569 - 0.821i$
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  + (−0.782 − 1.17i)2-s − 2.23·3-s + (−0.775 + 1.84i)4-s + (−0.983 − 2.00i)5-s + (1.74 + 2.63i)6-s + (3.76 + 1.22i)7-s + (2.77 − 0.529i)8-s + 1.98·9-s + (−1.59 + 2.72i)10-s + (−0.259 − 1.63i)11-s + (1.73 − 4.11i)12-s + (−5.87 + 1.90i)13-s + (−1.50 − 5.39i)14-s + (2.19 + 4.48i)15-s + (−2.79 − 2.85i)16-s + (0.889 − 1.22i)17-s + ⋯
L(s)  = 1  + (−0.553 − 0.832i)2-s − 1.28·3-s + (−0.387 + 0.921i)4-s + (−0.439 − 0.898i)5-s + (0.713 + 1.07i)6-s + (1.42 + 0.462i)7-s + (0.982 − 0.187i)8-s + 0.662·9-s + (−0.504 + 0.863i)10-s + (−0.0781 − 0.493i)11-s + (0.499 − 1.18i)12-s + (−1.62 + 0.529i)13-s + (−0.402 − 1.44i)14-s + (0.566 + 1.15i)15-s + (−0.699 − 0.714i)16-s + (0.215 − 0.296i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.569 - 0.821i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.569 - 0.821i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.245898 + 0.128781i\)
\(L(\frac12)\) \(\approx\) \(0.245898 + 0.128781i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.782 + 1.17i)T \)
5 \( 1 + (0.983 + 2.00i)T \)
41 \( 1 + (3.48 - 5.36i)T \)
good3 \( 1 + 2.23T + 3T^{2} \)
7 \( 1 + (-3.76 - 1.22i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.259 + 1.63i)T + (-10.4 + 3.39i)T^{2} \)
13 \( 1 + (5.87 - 1.90i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.889 + 1.22i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.09 - 1.57i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (-3.35 + 6.58i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (8.67 + 1.37i)T + (27.5 + 8.96i)T^{2} \)
31 \( 1 + (2.81 - 3.87i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-5.76 - 0.912i)T + (35.1 + 11.4i)T^{2} \)
43 \( 1 + (-4.87 - 2.48i)T + (25.2 + 34.7i)T^{2} \)
47 \( 1 + (-2.80 - 8.62i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.27 - 3.12i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.23 - 3.80i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.30 - 0.425i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (5.15 - 3.74i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-1.54 - 9.77i)T + (-67.5 + 21.9i)T^{2} \)
73 \( 1 + (-1.72 - 1.72i)T + 73iT^{2} \)
79 \( 1 + (-1.06 + 1.06i)T - 79iT^{2} \)
83 \( 1 + (0.379 + 0.379i)T + 83iT^{2} \)
89 \( 1 + (0.948 - 0.483i)T + (52.3 - 72.0i)T^{2} \)
97 \( 1 + (2.06 + 2.84i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.70129174283818769244381469275, −9.556748393500145812854492675265, −8.746745550992876546801108578836, −7.995611769238822336914550012826, −7.16107210416556656748033894626, −5.65410147694817442678807512857, −4.81070815104781507361729700649, −4.36483719410181032085522236440, −2.45296549912852334934731407133, −1.14974863838655743962397055906, 0.22495807956746229513686624879, 1.99173416265244999799755732376, 4.13874617131749650050464408276, 5.12710311546089377827354922907, 5.60130965154706961942578882595, 6.90773408939686875074080244185, 7.41010932663621872378749474523, 7.966670443322025564600159160531, 9.347398726557385689919087031852, 10.30211700483693857031849932648

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