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Properties

Label 2-2020-2020.1323-c0-0-0
Degree $2$
Conductor $2020$
Sign $-0.983 + 0.180i$
Analytic cond. $1.00811$
Root an. cond. $1.00404$
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  − 2-s + 4-s + i·5-s − 8-s − 9-s − i·10-s + (−1 + i)13-s + 16-s + (−1 − i)17-s + 18-s + i·20-s − 25-s + (1 − i)26-s + (−1 − i)29-s − 32-s + ⋯
L(s)  = 1  − 2-s + 4-s + i·5-s − 8-s − 9-s − i·10-s + (−1 + i)13-s + 16-s + (−1 − i)17-s + 18-s + i·20-s − 25-s + (1 − i)26-s + (−1 − i)29-s − 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.983 + 0.180i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.983 + 0.180i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1138756719\)
\(L(\frac12)\) \(\approx\) \(0.1138756719\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - iT \)
101 \( 1 - iT \)
good3 \( 1 + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (1 + i)T + iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + (1 + i)T + iT^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1 - i)T - iT^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−9.617711312510184687352955747308, −9.075770868735168549062319201054, −8.267621666132825087698101001072, −7.28948083927223337168992274593, −6.90534446519326496844169280284, −6.09323366888765455709934390709, −5.11331199430043144849976252754, −3.73921021074957728130033233653, −2.63915717187249021501897385292, −2.11751476826025078286040267395, 0.10142477622220214102835326810, 1.68042900576269386935768680825, 2.69636591441824511647457563129, 3.82771468143575295064728598790, 5.20992742452244129254956919887, 5.66961426651438699148558280432, 6.73087850221470863719087268336, 7.60917059722267281651755275613, 8.422302836974559515224150725955, 8.741783499603394472697486185604

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