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Properties

Label 2-820-205.68-c1-0-18
Degree $2$
Conductor $820$
Sign $-0.853 + 0.520i$
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  + (−1.03 − 0.427i)3-s + (0.661 − 2.13i)5-s + (1.80 − 4.35i)7-s + (−1.24 − 1.24i)9-s + (−0.166 − 0.401i)11-s + (0.389 + 0.161i)13-s + (−1.59 + 1.92i)15-s + (−2.80 + 1.16i)17-s + (−0.838 + 2.02i)19-s + (−3.72 + 3.72i)21-s + (2.60 − 2.60i)23-s + (−4.12 − 2.82i)25-s + (2.03 + 4.90i)27-s + (0.255 + 0.615i)29-s + 7.81i·31-s + ⋯
L(s)  = 1  + (−0.595 − 0.246i)3-s + (0.295 − 0.955i)5-s + (0.682 − 1.64i)7-s + (−0.413 − 0.413i)9-s + (−0.0501 − 0.121i)11-s + (0.107 + 0.0447i)13-s + (−0.411 + 0.495i)15-s + (−0.681 + 0.282i)17-s + (−0.192 + 0.464i)19-s + (−0.812 + 0.812i)21-s + (0.544 − 0.544i)23-s + (−0.825 − 0.564i)25-s + (0.390 + 0.943i)27-s + (0.0473 + 0.114i)29-s + 1.40i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.853 + 0.520i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.853 + 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290765 - 1.03473i\)
\(L(\frac12)\) \(\approx\) \(0.290765 - 1.03473i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.661 + 2.13i)T \)
41 \( 1 + (1.54 + 6.21i)T \)
good3 \( 1 + (1.03 + 0.427i)T + (2.12 + 2.12i)T^{2} \)
7 \( 1 + (-1.80 + 4.35i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (0.166 + 0.401i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.389 - 0.161i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + (2.80 - 1.16i)T + (12.0 - 12.0i)T^{2} \)
19 \( 1 + (0.838 - 2.02i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (-2.60 + 2.60i)T - 23iT^{2} \)
29 \( 1 + (-0.255 - 0.615i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 7.81iT - 31T^{2} \)
37 \( 1 + (3.42 - 3.42i)T - 37iT^{2} \)
43 \( 1 + 0.269iT - 43T^{2} \)
47 \( 1 + (-4.37 + 1.81i)T + (33.2 - 33.2i)T^{2} \)
53 \( 1 + (-1.93 + 4.68i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + 3.56iT - 59T^{2} \)
61 \( 1 + (2.86 + 2.86i)T + 61iT^{2} \)
67 \( 1 + (-12.6 + 5.24i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-0.878 + 0.363i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + 4.54iT - 73T^{2} \)
79 \( 1 + (12.8 - 5.32i)T + (55.8 - 55.8i)T^{2} \)
83 \( 1 + (2.66 + 2.66i)T + 83iT^{2} \)
89 \( 1 + (3.99 + 9.65i)T + (-62.9 + 62.9i)T^{2} \)
97 \( 1 + (-3.02 - 7.30i)T + (-68.5 + 68.5i)T^{2} \)
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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

−10.09054344885332356699222801549, −8.850737394518770591842845478635, −8.316463641529737235593211721648, −7.16680986800824343659880383633, −6.46571764667904087498034001540, −5.33118172725382379233662101442, −4.58717849071946689756130972518, −3.59585332348988759514174492024, −1.63362189674158147543806358688, −0.57378299803781610869544910804, 2.14066344061922920066905931172, 2.82855265269767454348202085628, 4.49089792228550508962771186728, 5.51510258879697902609972284986, 5.94178822754567562224358562926, 7.03788075678162455692572228143, 8.105796005524172779844520105773, 8.944161733289025602879354924246, 9.755492623235680569871348511850, 10.83671442490157793363752812930

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