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Properties

Label 2-820-41.23-c1-0-7
Degree $2$
Conductor $820$
Sign $0.528 - 0.848i$
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  + 1.65i·3-s + (−0.809 + 0.587i)5-s + (4.93 + 1.60i)7-s + 0.259·9-s + (1.15 − 1.58i)11-s + (4.57 − 1.48i)13-s + (−0.972 − 1.33i)15-s + (3.75 − 5.17i)17-s + (−7.06 − 2.29i)19-s + (−2.65 + 8.16i)21-s + (0.536 + 1.65i)23-s + (0.309 − 0.951i)25-s + 5.39i·27-s + (−2.01 − 2.76i)29-s + (3.97 + 2.89i)31-s + ⋯
L(s)  = 1  + 0.955i·3-s + (−0.361 + 0.262i)5-s + (1.86 + 0.605i)7-s + 0.0866·9-s + (0.348 − 0.479i)11-s + (1.26 − 0.412i)13-s + (−0.251 − 0.345i)15-s + (0.911 − 1.25i)17-s + (−1.62 − 0.526i)19-s + (−0.578 + 1.78i)21-s + (0.111 + 0.344i)23-s + (0.0618 − 0.190i)25-s + 1.03i·27-s + (−0.373 − 0.513i)29-s + (0.714 + 0.519i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.528 - 0.848i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.528 - 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71680 + 0.953463i\)
\(L(\frac12)\) \(\approx\) \(1.71680 + 0.953463i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-4.72 - 4.32i)T \)
good3 \( 1 - 1.65iT - 3T^{2} \)
7 \( 1 + (-4.93 - 1.60i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (-1.15 + 1.58i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (-4.57 + 1.48i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-3.75 + 5.17i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (7.06 + 2.29i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.536 - 1.65i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (2.01 + 2.76i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.97 - 2.89i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (5.19 - 3.77i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (0.898 + 2.76i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (6.40 - 2.08i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (6.09 + 8.39i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.37 + 10.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.834 - 2.56i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-5.27 - 7.25i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.13 - 7.07i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 - 8.06iT - 79T^{2} \)
83 \( 1 - 0.0283T + 83T^{2} \)
89 \( 1 + (12.2 + 3.96i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.60 + 7.71i)T + (-29.9 + 92.2i)T^{2} \)
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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

−10.52614702819588282922810129536, −9.533086529003234339449907823700, −8.483517042318065839317157636995, −8.210457095678878284332902883162, −6.96966793639021528282564806449, −5.72750982237774364223404440584, −4.88237571812859640702452804839, −4.16111708799172033339225879758, −3.02486777503500494578320143786, −1.44530053844319331969281212855, 1.34964529425487501317011655607, 1.78924292869790209755571150575, 3.97765165830659853301327727172, 4.44975597932252667534825933036, 5.84183759909272571916872021667, 6.73901686052766965478793930161, 7.75260032848866798673502048448, 8.140539892346997934985221537303, 8.892297401368093957298878337081, 10.47762668639651320181216478423

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