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Properties

Label 2-4020-67.29-c1-0-7
Degree $2$
Conductor $4020$
Sign $0.357 - 0.933i$
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 + (−1.96 − 3.39i)7-s + 9-s + (−0.165 − 0.286i)11-s + (−3.41 + 5.92i)13-s + 15-s + (1.72 − 2.99i)17-s + (−3.54 + 6.13i)19-s + (−1.96 − 3.39i)21-s + (−0.989 + 1.71i)23-s + 25-s + 27-s + (−2.31 − 4.01i)29-s + (3.25 + 5.63i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (−0.741 − 1.28i)7-s + 0.333·9-s + (−0.0497 − 0.0862i)11-s + (−0.948 + 1.64i)13-s + 0.258·15-s + (0.418 − 0.725i)17-s + (−0.812 + 1.40i)19-s + (−0.428 − 0.741i)21-s + (−0.206 + 0.357i)23-s + 0.200·25-s + 0.192·27-s + (−0.429 − 0.744i)29-s + (0.584 + 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.357 - 0.933i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.357 - 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.681060494\)
\(L(\frac12)\) \(\approx\) \(1.681060494\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.78 - 5.78i)T \)
good7 \( 1 + (1.96 + 3.39i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.165 + 0.286i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.41 - 5.92i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.72 + 2.99i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.54 - 6.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.989 - 1.71i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.31 + 4.01i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.25 - 5.63i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.28 - 3.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.72 - 6.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.35T + 43T^{2} \)
47 \( 1 + (-4.85 - 8.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 + 5.68T + 59T^{2} \)
61 \( 1 + (-1.97 + 3.42i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (1.85 + 3.20i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.79 + 4.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.12 - 7.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.83 - 15.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.96T + 89T^{2} \)
97 \( 1 + (-1.00 + 1.74i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.619144902663451242407246905243, −7.72858424278854035104648665648, −7.13595025811268336273942428164, −6.56850966256985065480590122673, −5.74045984309843282652588355634, −4.49220701212485849244550951418, −4.10521751400757880242238339521, −3.13179695423739689598840765329, −2.19677202670367108844008695556, −1.17471166536992639255971997326, 0.44875660457083159657214217330, 2.25179573017051600010282751345, 2.55975269577300311059858085578, 3.44424557645974305853768701001, 4.54726594894563228326096804408, 5.61372165559873729681350325781, 5.81381025105162492967726562785, 6.93210706365963092855462690378, 7.57317362449503419110049739089, 8.542944258840353124430777448168

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