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Properties

Label 2-4020-67.29-c1-0-29
Degree $2$
Conductor $4020$
Sign $0.996 + 0.0878i$
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 + (1.00 + 1.73i)7-s + 9-s + (0.449 + 0.777i)11-s + (1.97 − 3.41i)13-s + 15-s + (0.257 − 0.446i)17-s + (−0.443 + 0.768i)19-s + (1.00 + 1.73i)21-s + (3.81 − 6.60i)23-s + 25-s + 27-s + (−1.58 − 2.73i)29-s + (3.37 + 5.85i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (0.379 + 0.657i)7-s + 0.333·9-s + (0.135 + 0.234i)11-s + (0.546 − 0.946i)13-s + 0.258·15-s + (0.0625 − 0.108i)17-s + (−0.101 + 0.176i)19-s + (0.219 + 0.379i)21-s + (0.795 − 1.37i)23-s + 0.200·25-s + 0.192·27-s + (−0.293 − 0.508i)29-s + (0.606 + 1.05i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.996 + 0.0878i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.996 + 0.0878i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.985854639\)
\(L(\frac12)\) \(\approx\) \(2.985854639\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.15 + 3.98i)T \)
good7 \( 1 + (-1.00 - 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.449 - 0.777i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.97 + 3.41i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.257 + 0.446i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.443 - 0.768i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.81 + 6.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.58 + 2.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.37 - 5.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.59 + 4.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.35 - 4.07i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 8.28T + 43T^{2} \)
47 \( 1 + (2.65 + 4.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9.38T + 53T^{2} \)
59 \( 1 - 8.88T + 59T^{2} \)
61 \( 1 + (0.714 - 1.23i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-2.30 - 3.99i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.17 + 2.02i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.31 - 2.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.20 - 3.81i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.68T + 89T^{2} \)
97 \( 1 + (1.26 - 2.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.619527409589166516425515887100, −7.87227624794952177228857257775, −6.96098907060775185089386904920, −6.24777710594969185333777868836, −5.40636588555583390932731261417, −4.75638227415159867037661458195, −3.72124909539470326319774701326, −2.80805726204206847963542407982, −2.12015128485502320545718310921, −0.957157858302404050509337483446, 1.10438969471323353034665391279, 1.89082178096160895400577239146, 3.00674612966000542852301071847, 3.85738118531547130645171852265, 4.54482041070991551701303522736, 5.48063741879608716149790401931, 6.34042026545941897175905333797, 7.08709633498796058633712376257, 7.68631771790465918799903078709, 8.564780608752510256476193825694

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