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Properties

Label 2-820-820.107-c1-0-34
Degree $2$
Conductor $820$
Sign $-0.770 - 0.637i$
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  + (−1.26 + 0.642i)2-s + (1.17 − 1.61i)4-s + (−1.03 + 1.98i)5-s + (−0.442 + 2.79i)8-s + 3i·9-s + (0.0240 − 3.16i)10-s + (5.83 − 2.97i)13-s + (−1.23 − 3.80i)16-s + (−0.973 + 6.14i)17-s + (−1.92 − 3.78i)18-s + (2 + 4.00i)20-s + (−2.87 − 4.08i)25-s + (−5.44 + 7.49i)26-s + (−8.09 − 5.87i)29-s + (4.00 + 4.00i)32-s + ⋯
L(s)  = 1  + (−0.891 + 0.453i)2-s + (0.587 − 0.809i)4-s + (−0.460 + 0.887i)5-s + (−0.156 + 0.987i)8-s + i·9-s + (0.00759 − 0.999i)10-s + (1.61 − 0.824i)13-s + (−0.309 − 0.951i)16-s + (−0.235 + 1.49i)17-s + (−0.453 − 0.891i)18-s + (0.447 + 0.894i)20-s + (−0.575 − 0.817i)25-s + (−1.06 + 1.47i)26-s + (−1.50 − 1.09i)29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.770 - 0.637i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.770 - 0.637i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.251948 + 0.700013i\)
\(L(\frac12)\) \(\approx\) \(0.251948 + 0.700013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.26 - 0.642i)T \)
5 \( 1 + (1.03 - 1.98i)T \)
41 \( 1 + (5.99 - 2.25i)T \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 + (-4.11 - 5.66i)T^{2} \)
11 \( 1 + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-5.83 + 2.97i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.973 - 6.14i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (13.5 - 18.6i)T^{2} \)
29 \( 1 + (8.09 + 5.87i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.90 - 12.0i)T + (-35.1 + 11.4i)T^{2} \)
43 \( 1 + (25.2 - 34.7i)T^{2} \)
47 \( 1 + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-2.23 - 14.1i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.18 + 3.64i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (11.8 - 11.8i)T - 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (3.09 + 9.51i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.02 - 0.795i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.64842766498593656951650144838, −9.849858060949769627917678648459, −8.504315733144680075334915851208, −8.127879751026821148701785909425, −7.33883525866500399444624970310, −6.24187869700900448246307235531, −5.73876400323258543535257340742, −4.19329541334263830959017629142, −2.93211064892920881565128093372, −1.58406999697469679068943714785, 0.51277280589301483373833396966, 1.73894478823022643574852889811, 3.43750302314832882994527045876, 4.07128956800255139444639700906, 5.52540992123274721370884096806, 6.73062065369323059570985323042, 7.42149079566663528673978495363, 8.628038496888344254052781668932, 9.006262811760259636994062239010, 9.588633940329196941011251956646

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