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Properties

Label 2-4020-67.29-c1-0-41
Degree $2$
Conductor $4020$
Sign $0.540 + 0.841i$
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  + 3-s + 5-s + (0.983 + 1.70i)7-s + 9-s + (−0.830 − 1.43i)11-s + (1.04 − 1.80i)13-s + 15-s + (3.67 − 6.36i)17-s + (2.16 − 3.75i)19-s + (0.983 + 1.70i)21-s + (−0.0409 + 0.0709i)23-s + 25-s + 27-s + (−4.86 − 8.43i)29-s + (−2.11 − 3.66i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (0.371 + 0.643i)7-s + 0.333·9-s + (−0.250 − 0.433i)11-s + (0.289 − 0.501i)13-s + 0.258·15-s + (0.891 − 1.54i)17-s + (0.497 − 0.861i)19-s + (0.214 + 0.371i)21-s + (−0.00853 + 0.0147i)23-s + 0.200·25-s + 0.192·27-s + (−0.904 − 1.56i)29-s + (−0.380 − 0.658i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 + 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.540 + 0.841i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (3781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.540 + 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.623954681\)
\(L(\frac12)\) \(\approx\) \(2.623954681\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
67 \( 1 + (1.23 + 8.09i)T \)
good7 \( 1 + (-0.983 - 1.70i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.830 + 1.43i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.04 + 1.80i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.67 + 6.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.16 + 3.75i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0409 - 0.0709i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.86 + 8.43i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.11 + 3.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.24 - 7.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.62 + 4.54i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.536T + 43T^{2} \)
47 \( 1 + (-2.27 - 3.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.41T + 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + (-0.416 + 0.721i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-5.10 - 8.83i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.703 - 1.21i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.45 - 7.71i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.88 + 6.72i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (5.51 - 9.55i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.232549641515075342109601336663, −7.76387841678575731829125996325, −6.98846751256072643504447895842, −5.97156837496972067970537685086, −5.36234299588585179687718256582, −4.69380953833506575931555263854, −3.44380669916074658888634619423, −2.81055207784730481415187222455, −1.98037151287712869265226589581, −0.69012124698618938199398945151, 1.43728001762022652080689174046, 1.84645139805285114733998666214, 3.31070965841399334469348479111, 3.78157314846755239497958354770, 4.77560753521525233301828630346, 5.59153562456199937725814554565, 6.37210548963071422264708543956, 7.33873647402503525313991719005, 7.72486528501084785699663302368, 8.593045478292300342127724814845

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