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Properties

Label 2-4020-67.37-c1-0-40
Degree $2$
Conductor $4020$
Sign $0.221 + 0.975i$
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 + (2.29 − 3.97i)7-s + 9-s + (2.82 − 4.89i)11-s + (−2.64 − 4.57i)13-s + 15-s + (3.09 + 5.36i)17-s + (0.0519 + 0.0899i)19-s + (2.29 − 3.97i)21-s + (2.07 + 3.59i)23-s + 25-s + 27-s + (3.10 − 5.37i)29-s + (−2.19 + 3.79i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (0.866 − 1.50i)7-s + 0.333·9-s + (0.851 − 1.47i)11-s + (−0.732 − 1.26i)13-s + 0.258·15-s + (0.751 + 1.30i)17-s + (0.0119 + 0.0206i)19-s + (0.500 − 0.866i)21-s + (0.432 + 0.749i)23-s + 0.200·25-s + 0.192·27-s + (0.576 − 0.998i)29-s + (−0.393 + 0.682i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.221 + 0.975i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.221 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.117293258\)
\(L(\frac12)\) \(\approx\) \(3.117293258\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.91 + 6.54i)T \)
good7 \( 1 + (-2.29 + 3.97i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.82 + 4.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.64 + 4.57i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.09 - 5.36i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0519 - 0.0899i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.07 - 3.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.10 + 5.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.19 - 3.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.218 + 0.379i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.91 - 3.31i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 5.08T + 43T^{2} \)
47 \( 1 + (3.13 - 5.42i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.60T + 53T^{2} \)
59 \( 1 + 1.94T + 59T^{2} \)
61 \( 1 + (1.08 + 1.87i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (6.29 - 10.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.95 - 6.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.36 - 2.36i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.87 + 10.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + (1.26 + 2.18i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.105639399997356876266789932633, −7.80047853040902293757774812793, −6.95258917815088908588671910898, −6.03600561736982219954048767188, −5.34672552141719037728987178290, −4.33253528771448060629253295170, −3.61918092747054113077794155790, −2.93499981911926321715206985510, −1.50784569005890838565262210026, −0.880208790634699439033289154348, 1.53061660434619892176387477602, 2.18535873561282855864067687697, 2.81777733365942231647686855489, 4.21032786001002258305723238538, 4.87579685764390861763048751912, 5.41811738042350019358438579128, 6.60069322064149882678231995564, 7.11655832514020274085643884350, 7.86674886001338384453183026422, 8.966723985492846029894826086027

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