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Properties

Label 2-820-41.10-c1-0-6
Degree $2$
Conductor $820$
Sign $-0.186 + 0.982i$
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  − 3.30·3-s + (−0.309 − 0.951i)5-s + (−1.74 + 1.26i)7-s + 7.90·9-s + (−0.472 + 1.45i)11-s + (1.40 + 1.02i)13-s + (1.02 + 3.14i)15-s + (−1.12 + 3.46i)17-s + (−1.89 + 1.37i)19-s + (5.76 − 4.19i)21-s + (0.714 + 0.519i)23-s + (−0.809 + 0.587i)25-s − 16.2·27-s + (−2.14 − 6.59i)29-s + (3.00 − 9.25i)31-s + ⋯
L(s)  = 1  − 1.90·3-s + (−0.138 − 0.425i)5-s + (−0.660 + 0.479i)7-s + 2.63·9-s + (−0.142 + 0.438i)11-s + (0.390 + 0.283i)13-s + (0.263 + 0.810i)15-s + (−0.272 + 0.839i)17-s + (−0.433 + 0.315i)19-s + (1.25 − 0.914i)21-s + (0.148 + 0.108i)23-s + (−0.161 + 0.117i)25-s − 3.11·27-s + (−0.397 − 1.22i)29-s + (0.540 − 1.66i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.186 + 0.982i$
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: \((2,\ 820,\ (\ :1/2),\ -0.186 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227802 - 0.275033i\)
\(L(\frac12)\) \(\approx\) \(0.227802 - 0.275033i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.79 + 4.23i)T \)
good3 \( 1 + 3.30T + 3T^{2} \)
7 \( 1 + (1.74 - 1.26i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (0.472 - 1.45i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.40 - 1.02i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.12 - 3.46i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.89 - 1.37i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.714 - 0.519i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (2.14 + 6.59i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.00 + 9.25i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.114 + 0.352i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-0.227 - 0.165i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (2.21 + 1.60i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.53 + 10.8i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (4.12 + 2.99i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.69 + 1.22i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.29 + 7.07i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (0.299 - 0.921i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 5.21T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + 14.4T + 83T^{2} \)
89 \( 1 + (13.5 - 9.87i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (3.95 + 12.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.01969120371625493119682512167, −9.523377345767289335269503879794, −8.213497987252926624997100394520, −7.14710056485891969385344616667, −6.11610886419481995849331864146, −5.88628357507786176750801198250, −4.68273432345181241541222835789, −3.95678014647516852503810750115, −1.88488461350677779072209605216, −0.27717179558397683369273589391, 1.06446391558176659401106666687, 3.14776849742534144975268051803, 4.39032214426257219474393916141, 5.25000556013988090755143132513, 6.19971289542592203619519191308, 6.78856978256282872918000152921, 7.48169121828550302328079962089, 8.944633806056184187548472020429, 10.03715125350736388142297596935, 10.66444227904353613306003221413

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