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Properties

Label 2-820-820.79-c1-0-95
Degree $2$
Conductor $820$
Sign $-0.986 - 0.161i$
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 − i)2-s + 2i·4-s + (−0.707 − 2.12i)5-s + (2 − 2i)8-s + (−2.12 − 2.12i)9-s + (−1.41 + 2.82i)10-s + (2.70 − 6.53i)13-s − 4·16-s + (3.12 + 7.53i)17-s + 4.24i·18-s + (4.24 − 1.41i)20-s + (−3.99 + 3i)25-s + (−9.24 + 3.82i)26-s + (−9.94 − 4.12i)29-s + (4 + 4i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + i·4-s + (−0.316 − 0.948i)5-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.447 + 0.894i)10-s + (0.750 − 1.81i)13-s − 16-s + (0.757 + 1.82i)17-s + 0.999i·18-s + (0.948 − 0.316i)20-s + (−0.799 + 0.600i)25-s + (−1.81 + 0.750i)26-s + (−1.84 − 0.765i)29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.986 - 0.161i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.986 - 0.161i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0449842 + 0.554794i\)
\(L(\frac12)\) \(\approx\) \(0.0449842 + 0.554794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
5 \( 1 + (0.707 + 2.12i)T \)
41 \( 1 + (5 - 4i)T \)
good3 \( 1 + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (-2.70 + 6.53i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + (-3.12 - 7.53i)T + (-12.0 + 12.0i)T^{2} \)
19 \( 1 + (-13.4 + 13.4i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (9.94 + 4.12i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 7.07iT - 37T^{2} \)
43 \( 1 - 43iT^{2} \)
47 \( 1 + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (13.3 + 5.53i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + (-1 + i)T - 61iT^{2} \)
67 \( 1 + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.24 + 4.24i)T + 73iT^{2} \)
79 \( 1 + (55.8 + 55.8i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-10.1 - 4.19i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-12.5 + 5.19i)T + (68.5 - 68.5i)T^{2} \)
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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

−9.754316277066726498133719604753, −8.940905210511269011637845289626, −8.103051445841868760480756965802, −7.86308924091486569462354405431, −6.19063796034682818475558656604, −5.38394613941832508825305105956, −3.83853718245906082686026787274, −3.33968216282903685115868338427, −1.62166406360878672703992067504, −0.35014140510027979161256222762, 1.81880400826999813819066451839, 3.14967290578756440157427141341, 4.60539556236201893547162422777, 5.62104587519795432395626431725, 6.60290203001886536455832792613, 7.25749475498142304075922341956, 7.992539904489952718912734287048, 9.037227612110527517906374836279, 9.600133831025006584837537034552, 10.68460472893961800646836338648

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