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Properties

Label 2-820-820.583-c1-0-82
Degree $2$
Conductor $820$
Sign $0.933 + 0.357i$
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  + (−1.27 + 0.611i)2-s + 1.22·3-s + (1.25 − 1.55i)4-s + (1.86 + 1.23i)5-s + (−1.55 + 0.747i)6-s − 2.99i·7-s + (−0.643 + 2.75i)8-s − 1.50·9-s + (−3.13 − 0.435i)10-s + (−0.0849 + 0.0849i)11-s + (1.53 − 1.90i)12-s − 6.06i·13-s + (1.82 + 3.81i)14-s + (2.27 + 1.50i)15-s + (−0.862 − 3.90i)16-s + 6.38i·17-s + ⋯
L(s)  = 1  + (−0.901 + 0.432i)2-s + 0.705·3-s + (0.626 − 0.779i)4-s + (0.833 + 0.552i)5-s + (−0.636 + 0.305i)6-s − 1.13i·7-s + (−0.227 + 0.973i)8-s − 0.501·9-s + (−0.990 − 0.137i)10-s + (−0.0256 + 0.0256i)11-s + (0.441 − 0.550i)12-s − 1.68i·13-s + (0.488 + 1.01i)14-s + (0.588 + 0.389i)15-s + (−0.215 − 0.976i)16-s + 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.933 + 0.357i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.933 + 0.357i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40871 - 0.260264i\)
\(L(\frac12)\) \(\approx\) \(1.40871 - 0.260264i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.611i)T \)
5 \( 1 + (-1.86 - 1.23i)T \)
41 \( 1 + (2.83 - 5.74i)T \)
good3 \( 1 - 1.22T + 3T^{2} \)
7 \( 1 + 2.99iT - 7T^{2} \)
11 \( 1 + (0.0849 - 0.0849i)T - 11iT^{2} \)
13 \( 1 + 6.06iT - 13T^{2} \)
17 \( 1 - 6.38iT - 17T^{2} \)
19 \( 1 + (-5.18 + 5.18i)T - 19iT^{2} \)
23 \( 1 + (-5.57 + 5.57i)T - 23iT^{2} \)
29 \( 1 + (0.521 - 0.521i)T - 29iT^{2} \)
31 \( 1 + 1.55iT - 31T^{2} \)
37 \( 1 + (-4.51 + 4.51i)T - 37iT^{2} \)
43 \( 1 + (-0.300 - 0.300i)T + 43iT^{2} \)
47 \( 1 + 4.09T + 47T^{2} \)
53 \( 1 + 9.00iT - 53T^{2} \)
59 \( 1 + 4.12T + 59T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 - 6.34T + 67T^{2} \)
71 \( 1 + (2.90 - 2.90i)T - 71iT^{2} \)
73 \( 1 + (-0.672 + 0.672i)T - 73iT^{2} \)
79 \( 1 + (-9.02 - 9.02i)T + 79iT^{2} \)
83 \( 1 + (-7.89 + 7.89i)T - 83iT^{2} \)
89 \( 1 + (4.98 - 4.98i)T - 89iT^{2} \)
97 \( 1 + 7.96iT - 97T^{2} \)
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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

−10.12727273692234047173490229017, −9.330631099638103527789837709973, −8.431123966782653684196011233589, −7.73469571214968118532626320597, −6.93245852419195730349447615070, −6.02709965174364275524076363220, −5.11273018205915447298440234776, −3.33155427235586771523554991066, −2.50193213195145484479976243969, −0.934490985596740071728981020137, 1.52677410696503385520559306812, 2.46479894114571090614474975136, 3.35769402008893076756676442991, 4.99907233937827459960365161060, 5.96087102624457090932047844488, 7.07359191704428310758425773984, 8.045439818786376201297933106134, 8.989140332095392788314643894083, 9.310659445069387889902992900596, 9.731442976343555399856068856823

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