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Properties

Label 2-4020-67.37-c1-0-35
Degree $2$
Conductor $4020$
Sign $-0.566 + 0.824i$
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 + (−1.13 + 1.97i)7-s + 9-s + (1.34 − 2.32i)11-s + (−0.0777 − 0.134i)13-s − 15-s + (−0.386 − 0.669i)17-s + (−0.333 − 0.576i)19-s + (1.13 − 1.97i)21-s + (−2.24 − 3.88i)23-s + 25-s − 27-s + (0.790 − 1.36i)29-s + (−2.92 + 5.06i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (−0.430 + 0.745i)7-s + 0.333·9-s + (0.404 − 0.700i)11-s + (−0.0215 − 0.0373i)13-s − 0.258·15-s + (−0.0937 − 0.162i)17-s + (−0.0764 − 0.132i)19-s + (0.248 − 0.430i)21-s + (−0.467 − 0.809i)23-s + 0.200·25-s − 0.192·27-s + (0.146 − 0.254i)29-s + (−0.525 + 0.909i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.566 + 0.824i$
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: \((2,\ 4020,\ (\ :1/2),\ -0.566 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6399756355\)
\(L(\frac12)\) \(\approx\) \(0.6399756355\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 - 8.04i)T \)
good7 \( 1 + (1.13 - 1.97i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.34 + 2.32i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0777 + 0.134i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.386 + 0.669i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.333 + 0.576i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.24 + 3.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.790 + 1.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.92 - 5.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.60 + 7.98i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.88 - 8.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 4.88T + 43T^{2} \)
47 \( 1 + (-4.20 + 7.27i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.78T + 53T^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 + (-1.14 - 1.98i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-6.37 + 11.0i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.73 + 6.47i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.85 - 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.19 + 7.26i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 5.03T + 89T^{2} \)
97 \( 1 + (5.24 + 9.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.433868081718692736292133287423, −7.25969173113803916718578990742, −6.58869487715481073077547776178, −5.93547745365513215060521601968, −5.40516868967426238721006592704, −4.50930742038190656596547699138, −3.50092153464342225633422567759, −2.60834009416157604730123488888, −1.57213592032214889268937155692, −0.20383383770134357620869023909, 1.22319116109044082862061118669, 2.13612479685547947509275836944, 3.45826360972805800456698001748, 4.13555402103027563057948026337, 5.01229922272957985083299011869, 5.77169036291833640528898631687, 6.56321160186949905293736688905, 7.08444070283385028979621020759, 7.79402065823744273132850836259, 8.805680373281029043315124402840

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