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Properties

Label 2-820-820.439-c1-0-99
Degree $2$
Conductor $820$
Sign $-0.271 + 0.962i$
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  + (−0.221 + 1.39i)2-s + (1.16 − 2.81i)3-s + (−1.90 − 0.618i)4-s + (−1.99 + 1.01i)5-s + (3.66 + 2.24i)6-s + (2.31 − 1.41i)7-s + (1.28 − 2.52i)8-s + (−4.42 − 4.42i)9-s + (−0.977 − 3.00i)10-s + (−3.95 + 4.62i)12-s + (1.46 + 3.54i)14-s + (0.533 + 6.78i)15-s + (3.23 + 2.35i)16-s + (7.15 − 5.19i)18-s + (4.41 − 0.699i)20-s + (−1.28 − 8.13i)21-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)2-s + (0.672 − 1.62i)3-s + (−0.951 − 0.309i)4-s + (−0.891 + 0.453i)5-s + (1.49 + 0.917i)6-s + (0.873 − 0.535i)7-s + (0.453 − 0.891i)8-s + (−1.47 − 1.47i)9-s + (−0.309 − 0.951i)10-s + (−1.14 + 1.33i)12-s + (0.391 + 0.946i)14-s + (0.137 + 1.75i)15-s + (0.809 + 0.587i)16-s + (1.68 − 1.22i)18-s + (0.987 − 0.156i)20-s + (−0.281 − 1.77i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 820 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.271 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(820\)    =    \(2^{2} \cdot 5 \cdot 41\)
Sign: $-0.271 + 0.962i$
Analytic conductor: \(6.54773\)
Root analytic conductor: \(2.55885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{820} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 820,\ (\ :1/2),\ -0.271 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.647794 - 0.855753i\)
\(L(\frac12)\) \(\approx\) \(0.647794 - 0.855753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.221 - 1.39i)T \)
5 \( 1 + (1.99 - 1.01i)T \)
41 \( 1 + (-1.26 + 6.27i)T \)
good3 \( 1 + (-1.16 + 2.81i)T + (-2.12 - 2.12i)T^{2} \)
7 \( 1 + (-2.31 + 1.41i)T + (3.17 - 6.23i)T^{2} \)
11 \( 1 + (10.8 + 1.72i)T^{2} \)
13 \( 1 + (11.5 - 5.90i)T^{2} \)
17 \( 1 + (-2.65 + 16.7i)T^{2} \)
19 \( 1 + (16.9 + 8.62i)T^{2} \)
23 \( 1 + (4.02 + 5.53i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (8.13 + 6.94i)T + (4.53 + 28.6i)T^{2} \)
31 \( 1 + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (29.9 + 21.7i)T^{2} \)
43 \( 1 + (1.99 - 12.6i)T + (-40.8 - 13.2i)T^{2} \)
47 \( 1 + (11.3 + 6.92i)T + (21.3 + 41.8i)T^{2} \)
53 \( 1 + (8.29 + 52.3i)T^{2} \)
59 \( 1 + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.39 - 0.221i)T + (58.0 - 18.8i)T^{2} \)
67 \( 1 + (-13.4 + 1.05i)T + (66.1 - 10.4i)T^{2} \)
71 \( 1 + (-70.1 - 11.1i)T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + (-55.8 - 55.8i)T^{2} \)
83 \( 1 - 17.7T + 83T^{2} \)
89 \( 1 + (-8.53 + 5.23i)T + (40.4 - 79.2i)T^{2} \)
97 \( 1 + (95.8 - 15.1i)T^{2} \)
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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

−9.704384744772493521839481938782, −8.499142492977477771933434436430, −8.019321706472154159019048141664, −7.55841361307867667829990159291, −6.79429155873533881932662935610, −6.05832258616761030088696420036, −4.60033507951228715627006309665, −3.54990194560283119423675755771, −1.99579492744385707912219986073, −0.52045180091402228627542089386, 1.90898962162832668639027754473, 3.34349965656125609421890469887, 3.86624936620068747996301924285, 4.89329541377054053481657308070, 5.32925587328779875765202371846, 7.70783877098124500795816767012, 8.285277414213660254976468464233, 9.032517476655709337715235270602, 9.543779774969243385893661529060, 10.50989368931055726305852495536

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