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Properties

Label 2-820-41.10-c1-0-5
Degree $2$
Conductor $820$
Sign $0.632 - 0.774i$
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  + 0.124·3-s + (0.309 + 0.951i)5-s + (2.19 − 1.59i)7-s − 2.98·9-s + (−1.48 + 4.56i)11-s + (4.36 + 3.16i)13-s + (0.0384 + 0.118i)15-s + (0.608 − 1.87i)17-s + (−0.677 + 0.491i)19-s + (0.272 − 0.197i)21-s + (3.56 + 2.58i)23-s + (−0.809 + 0.587i)25-s − 0.743·27-s + (2.94 + 9.06i)29-s + (1.67 − 5.15i)31-s + ⋯
L(s)  = 1  + 0.0717·3-s + (0.138 + 0.425i)5-s + (0.828 − 0.601i)7-s − 0.994·9-s + (−0.447 + 1.37i)11-s + (1.20 + 0.878i)13-s + (0.00991 + 0.0305i)15-s + (0.147 − 0.454i)17-s + (−0.155 + 0.112i)19-s + (0.0594 − 0.0431i)21-s + (0.743 + 0.539i)23-s + (−0.161 + 0.117i)25-s − 0.143·27-s + (0.546 + 1.68i)29-s + (0.300 − 0.926i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.632 - 0.774i$
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: \((2,\ 820,\ (\ :1/2),\ 0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47956 + 0.701483i\)
\(L(\frac12)\) \(\approx\) \(1.47956 + 0.701483i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.24 + 1.43i)T \)
good3 \( 1 - 0.124T + 3T^{2} \)
7 \( 1 + (-2.19 + 1.59i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.48 - 4.56i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-4.36 - 3.16i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.608 + 1.87i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.677 - 0.491i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-3.56 - 2.58i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.94 - 9.06i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.67 + 5.15i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.25 + 3.85i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (3.61 + 2.62i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-5.60 - 4.06i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.56 - 10.9i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.41 + 3.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (3.17 - 2.30i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.76 - 11.5i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-4.61 + 14.1i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + (1.88 - 1.36i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.41 + 13.5i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.60863383203722221749965272266, −9.410752618283750289592472234692, −8.732355112678161167729890609373, −7.66080092272938515369915046537, −7.07003101331921909036506037944, −5.97795040412570169830422506876, −4.95188255626363373882509097463, −4.02805588097716046628214726099, −2.75301073897280034392932092507, −1.50882915415705422386226949811, 0.885999942713484919737862029519, 2.53695730123990531312208575712, 3.50730977046206379372199151057, 4.95095366056159537604615162496, 5.71686606777818492434197892724, 6.30707488577499845358261620009, 8.098914709146661843185347932952, 8.345418979293245118031046838791, 8.903158381534493287740764744539, 10.25991723338903357695215922461

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