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Properties

Label 2-820-820.147-c0-0-0
Degree $2$
Conductor $820$
Sign $0.988 - 0.151i$
Analytic cond. $0.409233$
Root an. cond. $0.639713$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.987 + 0.156i)5-s + (0.951 + 0.309i)8-s + (−0.707 − 0.707i)9-s + (0.453 − 0.891i)10-s + (1.29 − 0.794i)13-s + (−0.809 + 0.587i)16-s + (1.70 − 0.133i)17-s + (0.987 − 0.156i)18-s + (0.453 + 0.891i)20-s + (0.951 − 0.309i)25-s + (−0.119 + 1.51i)26-s + (−0.581 + 0.497i)29-s − i·32-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.309 − 0.951i)4-s + (−0.987 + 0.156i)5-s + (0.951 + 0.309i)8-s + (−0.707 − 0.707i)9-s + (0.453 − 0.891i)10-s + (1.29 − 0.794i)13-s + (−0.809 + 0.587i)16-s + (1.70 − 0.133i)17-s + (0.987 − 0.156i)18-s + (0.453 + 0.891i)20-s + (0.951 − 0.309i)25-s + (−0.119 + 1.51i)26-s + (−0.581 + 0.497i)29-s − i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $0.988 - 0.151i$
Analytic conductor: \(0.409233\)
Root analytic conductor: \(0.639713\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :0),\ 0.988 - 0.151i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6064736304\)
\(L(\frac12)\) \(\approx\) \(0.6064736304\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 - 0.809i)T \)
5 \( 1 + (0.987 - 0.156i)T \)
41 \( 1 + (-0.587 + 0.809i)T \)
good3 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.891 + 0.453i)T^{2} \)
11 \( 1 + (-0.987 + 0.156i)T^{2} \)
13 \( 1 + (-1.29 + 0.794i)T + (0.453 - 0.891i)T^{2} \)
17 \( 1 + (-1.70 + 0.133i)T + (0.987 - 0.156i)T^{2} \)
19 \( 1 + (0.891 - 0.453i)T^{2} \)
23 \( 1 + (0.951 - 0.309i)T^{2} \)
29 \( 1 + (0.581 - 0.497i)T + (0.156 - 0.987i)T^{2} \)
31 \( 1 + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.863 + 1.69i)T + (-0.587 - 0.809i)T^{2} \)
43 \( 1 + (-0.309 - 0.951i)T^{2} \)
47 \( 1 + (0.891 + 0.453i)T^{2} \)
53 \( 1 + (0.152 - 1.93i)T + (-0.987 - 0.156i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (1.76 + 0.278i)T + (0.951 + 0.309i)T^{2} \)
67 \( 1 + (-0.156 + 0.987i)T^{2} \)
71 \( 1 + (0.987 - 0.156i)T^{2} \)
73 \( 1 + 0.907T + T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1.57 + 0.965i)T + (0.453 + 0.891i)T^{2} \)
97 \( 1 + (-1.29 - 1.51i)T + (-0.156 + 0.987i)T^{2} \)
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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

−10.52495393866597806939934656473, −9.311574858748106854382672558292, −8.689506914774628985646565142445, −7.82572566548078833585136801589, −7.30343177504427440026286540612, −6.02142207398541117420818271076, −5.57489541355958420240297106564, −4.08758903008075660352610827596, −3.15869030708692286571206054275, −0.924202550874431193811352940337, 1.34877257796171283199254093700, 2.94166812989547524497704088524, 3.78564457964779567225930800430, 4.76679343694706103386557715884, 6.07251220292159488629286954261, 7.41797430465354777025367140305, 8.099990337131782739599924919075, 8.626283205868212537863213122659, 9.614701527257717798934456794834, 10.53400379384113810088499895927

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