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Properties

Label 2-820-41.37-c1-0-2
Degree $2$
Conductor $820$
Sign $-0.511 - 0.859i$
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  − 0.279·3-s + (−0.309 + 0.951i)5-s + (3.56 + 2.58i)7-s − 2.92·9-s + (1.33 + 4.12i)11-s + (−3.53 + 2.56i)13-s + (0.0864 − 0.266i)15-s + (−1.00 − 3.08i)17-s + (−6.74 − 4.89i)19-s + (−0.997 − 0.724i)21-s + (0.252 − 0.183i)23-s + (−0.809 − 0.587i)25-s + 1.65·27-s + (−0.305 + 0.941i)29-s + (1.54 + 4.76i)31-s + ⋯
L(s)  = 1  − 0.161·3-s + (−0.138 + 0.425i)5-s + (1.34 + 0.978i)7-s − 0.973·9-s + (0.403 + 1.24i)11-s + (−0.979 + 0.711i)13-s + (0.0223 − 0.0687i)15-s + (−0.242 − 0.747i)17-s + (−1.54 − 1.12i)19-s + (−0.217 − 0.158i)21-s + (0.0526 − 0.0382i)23-s + (−0.161 − 0.117i)25-s + 0.318·27-s + (−0.0567 + 0.174i)29-s + (0.277 + 0.855i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.511 - 0.859i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.511 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.538274 + 0.946442i\)
\(L(\frac12)\) \(\approx\) \(0.538274 + 0.946442i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.96 - 2.34i)T \)
good3 \( 1 + 0.279T + 3T^{2} \)
7 \( 1 + (-3.56 - 2.58i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-1.33 - 4.12i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.53 - 2.56i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.00 + 3.08i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (6.74 + 4.89i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.252 + 0.183i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.305 - 0.941i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.54 - 4.76i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.26 - 6.97i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-5.98 + 4.34i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (8.75 - 6.36i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.80 - 5.56i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.68 - 6.30i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.57 + 3.32i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-4.64 + 14.2i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.51 - 7.74i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 - 5.48T + 79T^{2} \)
83 \( 1 + 11.4T + 83T^{2} \)
89 \( 1 + (-6.27 - 4.55i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.42 + 4.37i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.79116425847117148486527670749, −9.466284117018274048962388288335, −8.899369363938882852519254920204, −8.001638281301903582322652574154, −7.04631542281353751073906536034, −6.24230727468101351143562966687, −4.82428800206695757564624616062, −4.68567091338674974645484343444, −2.71480213680097168440574955504, −1.99514268967562369393561917879, 0.52571916551186304388163355890, 2.04296092132661369329093792421, 3.63869915184627385279612113396, 4.50272583919746975076077974596, 5.53029009299764267163649426783, 6.28795448852059767244970338700, 7.71859634805554668044940423389, 8.171934796777374883238328836612, 8.847101663583569962220221741175, 10.15730627118149812915476199973

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