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Properties

Label 2-2020-2020.1303-c0-0-0
Degree $2$
Conductor $2020$
Sign $-0.277 - 0.960i$
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  + i·2-s − 4-s + i·5-s − i·8-s + 9-s − 10-s + (1 − i)13-s + 16-s + (1 + i)17-s + i·18-s − i·20-s − 25-s + (1 + i)26-s + (1 − i)29-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + i·5-s − i·8-s + 9-s − 10-s + (1 − i)13-s + 16-s + (1 + i)17-s + i·18-s − i·20-s − 25-s + (1 + i)26-s + (1 − i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2020\)    =    \(2^{2} \cdot 5 \cdot 101\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(1.00811\)
Root analytic conductor: \(1.00404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2020} (1303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2020,\ (\ :0),\ -0.277 - 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.188797259\)
\(L(\frac12)\) \(\approx\) \(1.188797259\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
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 \( 1 + 2T + T^{2} \)
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

−9.725385269557706069257217287555, −8.305268524995034015236058714125, −8.097304958777544708116983187874, −7.14188854016626049583539122842, −6.42694480738128296646109326158, −5.88631697316663989071020696742, −4.85062613600162812452170091523, −3.80240539834238719712248279858, −3.18104401496849088629683708978, −1.39149118860042497833945647567, 1.09745480967196114653113776514, 1.86257876700059487675505096884, 3.32540424364495147552428112965, 4.11705558750730057855754855488, 4.86917798788464873872750683480, 5.58233088051424932425939707550, 6.82830771545405395512903653069, 7.72938144801810746818231253643, 8.773334931064270371810945443470, 9.022591624788946396033860918397

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