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Properties

Label 2-820-20.3-c1-0-23
Degree $2$
Conductor $820$
Sign $0.733 - 0.679i$
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  + (−1.20 − 0.732i)2-s + (0.993 − 0.993i)3-s + (0.925 + 1.77i)4-s + (−1.63 − 1.52i)5-s + (−1.93 + 0.473i)6-s + (2.18 + 2.18i)7-s + (0.179 − 2.82i)8-s + 1.02i·9-s + (0.859 + 3.04i)10-s + 2.71i·11-s + (2.68 + 0.841i)12-s + (2.18 + 2.18i)13-s + (−1.04 − 4.24i)14-s + (−3.14 + 0.108i)15-s + (−2.28 + 3.28i)16-s + (−2.30 + 2.30i)17-s + ⋯
L(s)  = 1  + (−0.855 − 0.518i)2-s + (0.573 − 0.573i)3-s + (0.462 + 0.886i)4-s + (−0.731 − 0.682i)5-s + (−0.787 + 0.193i)6-s + (0.825 + 0.825i)7-s + (0.0634 − 0.997i)8-s + 0.341i·9-s + (0.271 + 0.962i)10-s + 0.819i·11-s + (0.774 + 0.242i)12-s + (0.605 + 0.605i)13-s + (−0.278 − 1.13i)14-s + (−0.810 + 0.0279i)15-s + (−0.571 + 0.820i)16-s + (−0.558 + 0.558i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.733 - 0.679i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ 0.733 - 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.839214 + 0.329185i\)
\(L(\frac12)\) \(\approx\) \(0.839214 + 0.329185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.20 + 0.732i)T \)
5 \( 1 + (1.63 + 1.52i)T \)
41 \( 1 - T \)
good3 \( 1 + (-0.993 + 0.993i)T - 3iT^{2} \)
7 \( 1 + (-2.18 - 2.18i)T + 7iT^{2} \)
11 \( 1 - 2.71iT - 11T^{2} \)
13 \( 1 + (-2.18 - 2.18i)T + 13iT^{2} \)
17 \( 1 + (2.30 - 2.30i)T - 17iT^{2} \)
19 \( 1 + 6.76T + 19T^{2} \)
23 \( 1 + (6.72 - 6.72i)T - 23iT^{2} \)
29 \( 1 + 8.06iT - 29T^{2} \)
31 \( 1 - 6.13iT - 31T^{2} \)
37 \( 1 + (1.43 - 1.43i)T - 37iT^{2} \)
43 \( 1 + (-2.56 + 2.56i)T - 43iT^{2} \)
47 \( 1 + (-4.33 - 4.33i)T + 47iT^{2} \)
53 \( 1 + (2.53 + 2.53i)T + 53iT^{2} \)
59 \( 1 - 0.274T + 59T^{2} \)
61 \( 1 + 2.40T + 61T^{2} \)
67 \( 1 + (-4.52 - 4.52i)T + 67iT^{2} \)
71 \( 1 + 3.75iT - 71T^{2} \)
73 \( 1 + (2.07 + 2.07i)T + 73iT^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + (-2.57 + 2.57i)T - 83iT^{2} \)
89 \( 1 + 12.7iT - 89T^{2} \)
97 \( 1 + (-2.97 + 2.97i)T - 97iT^{2} \)
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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

−10.33702355816337363044209110160, −9.155669037905624941909745127731, −8.574166907366449828967106522073, −8.048140028316189077109128411114, −7.39598461672223054059711686992, −6.19888542342142827183654126611, −4.67882351343109294278712364505, −3.83208174407067222188118575729, −2.15366443628046210856442407264, −1.71421284167166727202236468586, 0.54111121509767129065757272160, 2.44593090497753049600571672560, 3.75991951469996385511222725704, 4.55986766586049887165450654939, 6.09705764673395121296689948330, 6.78957396332233898014574841030, 7.88517942766319088938686983537, 8.358057712820618215093708242213, 9.039300425020631462055432805672, 10.28328033885255695685356141928

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