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Properties

Label 2-820-41.10-c1-0-11
Degree $2$
Conductor $820$
Sign $0.983 + 0.180i$
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  + 2.53·3-s + (0.309 + 0.951i)5-s + (2.92 − 2.12i)7-s + 3.43·9-s + (0.677 − 2.08i)11-s + (−0.433 − 0.314i)13-s + (0.784 + 2.41i)15-s + (0.573 − 1.76i)17-s + (−3.44 + 2.50i)19-s + (7.42 − 5.39i)21-s + (−0.406 − 0.295i)23-s + (−0.809 + 0.587i)25-s + 1.10·27-s + (−1.25 − 3.85i)29-s + (−2.97 + 9.14i)31-s + ⋯
L(s)  = 1  + 1.46·3-s + (0.138 + 0.425i)5-s + (1.10 − 0.803i)7-s + 1.14·9-s + (0.204 − 0.629i)11-s + (−0.120 − 0.0872i)13-s + (0.202 + 0.623i)15-s + (0.139 − 0.428i)17-s + (−0.790 + 0.574i)19-s + (1.61 − 1.17i)21-s + (−0.0847 − 0.0615i)23-s + (−0.161 + 0.117i)25-s + 0.213·27-s + (−0.232 − 0.715i)29-s + (−0.533 + 1.64i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.983 + 0.180i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.983 + 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.80239 - 0.255331i\)
\(L(\frac12)\) \(\approx\) \(2.80239 - 0.255331i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-4.26 - 4.77i)T \)
good3 \( 1 - 2.53T + 3T^{2} \)
7 \( 1 + (-2.92 + 2.12i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-0.677 + 2.08i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.433 + 0.314i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.573 + 1.76i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.44 - 2.50i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.406 + 0.295i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.25 + 3.85i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.97 - 9.14i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.08 + 3.32i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-0.275 - 0.200i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-0.751 - 0.546i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.97 - 9.14i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-8.01 - 5.82i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.28 - 2.38i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.46 + 13.7i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (3.10 - 9.57i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 1.59T + 73T^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 - 7.14T + 83T^{2} \)
89 \( 1 + (-3.09 + 2.24i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.83 + 8.71i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.22593847692295471009965784365, −9.165101102743457670677976218173, −8.480988174605266116916068199559, −7.74874037886303568532832564600, −7.13368791148285138205567219552, −5.85112003028161577287351204259, −4.50891895723484647270802937876, −3.66899708660760838810807869353, −2.65040057697340473738551647492, −1.49515296306016025839455911876, 1.79696175938556410072716059327, 2.40873011142100309007978183826, 3.82352611803082983535143579128, 4.72044599695298135947719184320, 5.74604542057780168069907891764, 7.11453120622096240585108801623, 7.978208864176493120133625684056, 8.610754071300840466509177499983, 9.149016948338814128617019919805, 9.960550274217809661720099469943

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