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Properties

Label 2-4020-5.4-c1-0-53
Degree $2$
Conductor $4020$
Sign $-0.999 + 0.0238i$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
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  − i·3-s + (−2.23 + 0.0533i)5-s − 4.50i·7-s − 9-s + 0.971·11-s + 1.87i·13-s + (0.0533 + 2.23i)15-s − 5.03i·17-s + 1.77·19-s − 4.50·21-s − 5.44i·23-s + (4.99 − 0.238i)25-s + i·27-s + 3.85·29-s + 5.05·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.999 + 0.0238i)5-s − 1.70i·7-s − 0.333·9-s + 0.292·11-s + 0.519i·13-s + (0.0137 + 0.577i)15-s − 1.21i·17-s + 0.407·19-s − 0.983·21-s − 1.13i·23-s + (0.998 − 0.0477i)25-s + 0.192i·27-s + 0.715·29-s + 0.907·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.999 + 0.0238i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (1609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -0.999 + 0.0238i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.151960322\)
\(L(\frac12)\) \(\approx\) \(1.151960322\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + iT \)
5 \( 1 + (2.23 - 0.0533i)T \)
67 \( 1 + iT \)
good7 \( 1 + 4.50iT - 7T^{2} \)
11 \( 1 - 0.971T + 11T^{2} \)
13 \( 1 - 1.87iT - 13T^{2} \)
17 \( 1 + 5.03iT - 17T^{2} \)
19 \( 1 - 1.77T + 19T^{2} \)
23 \( 1 + 5.44iT - 23T^{2} \)
29 \( 1 - 3.85T + 29T^{2} \)
31 \( 1 - 5.05T + 31T^{2} \)
37 \( 1 + 10.1iT - 37T^{2} \)
41 \( 1 - 5.79T + 41T^{2} \)
43 \( 1 + 8.65iT - 43T^{2} \)
47 \( 1 - 1.27iT - 47T^{2} \)
53 \( 1 - 4.85iT - 53T^{2} \)
59 \( 1 - 0.801T + 59T^{2} \)
61 \( 1 + 9.97T + 61T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + 9.73iT - 73T^{2} \)
79 \( 1 - 5.68T + 79T^{2} \)
83 \( 1 - 16.8iT - 83T^{2} \)
89 \( 1 + 16.4T + 89T^{2} \)
97 \( 1 + 11.3iT - 97T^{2} \)
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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

−7.85005705103318610642126506187, −7.23467889221038453880464376567, −6.99145553273647535855593196476, −6.09158293623378268889074323434, −4.76082270275259615024077108380, −4.31272636961553581900640899676, −3.50003389635570127384856585933, −2.54840865881914459902347325652, −1.07329176963190896397977778097, −0.40310591457020646475558901802, 1.40468821586827398367136628100, 2.80035285274793040042100298575, 3.28213919538464031719268666191, 4.32030346100750131776520579132, 5.00854852376213523787368158026, 5.85126852144442796215940688747, 6.38865361833447126871778561834, 7.56184573534612371492340186870, 8.330127226682971899281570458145, 8.591566691077703156193129409823

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