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Properties

Label 2-4020-67.37-c1-0-32
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: $\chi_{4020} (841, \cdot )$
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

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

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