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Properties

Label 2-4020-5.4-c1-0-59
Degree $2$
Conductor $4020$
Sign $-0.988 + 0.151i$
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 + (2.21 − 0.339i)5-s − 3.30i·7-s − 9-s − 4.99·11-s − 3.79i·13-s + (−0.339 − 2.21i)15-s − 1.38i·17-s + 0.658·19-s − 3.30·21-s + 2.41i·23-s + (4.77 − 1.49i)25-s + i·27-s + 1.08·29-s + 10.3·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.988 − 0.151i)5-s − 1.24i·7-s − 0.333·9-s − 1.50·11-s − 1.05i·13-s + (−0.0875 − 0.570i)15-s − 0.334i·17-s + 0.151·19-s − 0.720·21-s + 0.503i·23-s + (0.954 − 0.299i)25-s + 0.192i·27-s + 0.202·29-s + 1.86·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.988 + 0.151i$
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: \((2,\ 4020,\ (\ :1/2),\ -0.988 + 0.151i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.450031398\)
\(L(\frac12)\) \(\approx\) \(1.450031398\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 + 0.339i)T \)
67 \( 1 - iT \)
good7 \( 1 + 3.30iT - 7T^{2} \)
11 \( 1 + 4.99T + 11T^{2} \)
13 \( 1 + 3.79iT - 13T^{2} \)
17 \( 1 + 1.38iT - 17T^{2} \)
19 \( 1 - 0.658T + 19T^{2} \)
23 \( 1 - 2.41iT - 23T^{2} \)
29 \( 1 - 1.08T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + 5.40T + 41T^{2} \)
43 \( 1 + 7.60iT - 43T^{2} \)
47 \( 1 - 9.10iT - 47T^{2} \)
53 \( 1 + 10.7iT - 53T^{2} \)
59 \( 1 + 10.7T + 59T^{2} \)
61 \( 1 + 1.92T + 61T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 + 16.2T + 79T^{2} \)
83 \( 1 + 3.77iT - 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 - 2.40iT - 97T^{2} \)
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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

−7.957093949516503698693400597480, −7.41915583522717165061969541695, −6.71892903392386147175261400151, −5.79248806387905413026593947663, −5.26690212226350029153302864845, −4.43318562903811703678217229005, −3.15044074315198221111233902183, −2.53683335857577532883178468785, −1.35271023370703109945426603169, −0.39627026574417116352623434547, 1.65006803077820563525111098386, 2.62733192073754972482558647670, 3.01564324349661750767788780578, 4.62067841108520684217905330357, 4.93340053818278388460072812705, 5.98510303921371304589621770138, 6.20559798804020049479212083218, 7.31693095925664708744122180323, 8.422626880543269941780374416492, 8.695522584384092383403355469539

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