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Properties

Label 2-4020-201.200-c1-0-83
Degree $2$
Conductor $4020$
Sign $-0.990 - 0.139i$
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  + (0.293 − 1.70i)3-s − 5-s − 1.30i·7-s + (−2.82 − 1.00i)9-s − 0.0724·11-s − 4.54i·13-s + (−0.293 + 1.70i)15-s − 5.16i·17-s + 5.55·19-s + (−2.23 − 0.383i)21-s − 9.12i·23-s + 25-s + (−2.53 + 4.53i)27-s − 9.23i·29-s + 10.8i·31-s + ⋯
L(s)  = 1  + (0.169 − 0.985i)3-s − 0.447·5-s − 0.494i·7-s + (−0.942 − 0.333i)9-s − 0.0218·11-s − 1.25i·13-s + (−0.0757 + 0.440i)15-s − 1.25i·17-s + 1.27·19-s + (−0.487 − 0.0836i)21-s − 1.90i·23-s + 0.200·25-s + (−0.488 + 0.872i)27-s − 1.71i·29-s + 1.94i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.990 - 0.139i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -0.990 - 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.329159409\)
\(L(\frac12)\) \(\approx\) \(1.329159409\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.293 + 1.70i)T \)
5 \( 1 + T \)
67 \( 1 + (0.243 + 8.18i)T \)
good7 \( 1 + 1.30iT - 7T^{2} \)
11 \( 1 + 0.0724T + 11T^{2} \)
13 \( 1 + 4.54iT - 13T^{2} \)
17 \( 1 + 5.16iT - 17T^{2} \)
19 \( 1 - 5.55T + 19T^{2} \)
23 \( 1 + 9.12iT - 23T^{2} \)
29 \( 1 + 9.23iT - 29T^{2} \)
31 \( 1 - 10.8iT - 31T^{2} \)
37 \( 1 + 0.716T + 37T^{2} \)
41 \( 1 - 7.26T + 41T^{2} \)
43 \( 1 - 3.55iT - 43T^{2} \)
47 \( 1 - 5.42iT - 47T^{2} \)
53 \( 1 + 2.69T + 53T^{2} \)
59 \( 1 - 6.20iT - 59T^{2} \)
61 \( 1 - 11.4iT - 61T^{2} \)
71 \( 1 + 5.55iT - 71T^{2} \)
73 \( 1 + 6.69T + 73T^{2} \)
79 \( 1 + 8.46iT - 79T^{2} \)
83 \( 1 + 9.48iT - 83T^{2} \)
89 \( 1 - 6.51iT - 89T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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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

−7.79518803000669878127218557324, −7.52038047066850391032079599263, −6.74794382723718380293808510119, −5.95362335294127004288458947478, −5.12193600911063251239911856451, −4.25526466470274177684137843369, −3.00324789300284029150949689552, −2.72162764384067180057704619075, −1.13354768924287681652310645885, −0.41734662502931533994797233597, 1.54540157757158100165077268261, 2.64820208709812618508614312351, 3.76884946927950515464043043452, 3.96316058325953898300026878734, 5.20581161959384070785335960302, 5.58376206296714882975628801706, 6.59458731279633614515343611138, 7.53627564928320991847927712463, 8.139694145195110162202514779383, 9.095506699408496369974016053630

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