Properties

Label 2-4020-67.29-c1-0-25
Degree $2$
Conductor $4020$
Sign $0.999 + 0.0122i$
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 + (−1.09 − 1.89i)7-s + 9-s + (2.56 + 4.44i)11-s + (3.21 − 5.57i)13-s + 15-s + (0.653 − 1.13i)17-s + (−2.76 + 4.78i)19-s + (−1.09 − 1.89i)21-s + (−1.52 + 2.64i)23-s + 25-s + 27-s + (3.69 + 6.39i)29-s + (1.00 + 1.73i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (−0.413 − 0.716i)7-s + 0.333·9-s + (0.773 + 1.33i)11-s + (0.892 − 1.54i)13-s + 0.258·15-s + (0.158 − 0.274i)17-s + (−0.634 + 1.09i)19-s + (−0.238 − 0.413i)21-s + (−0.318 + 0.552i)23-s + 0.200·25-s + 0.192·27-s + (0.685 + 1.18i)29-s + (0.179 + 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.999 + 0.0122i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.999 + 0.0122i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.799197947\)
\(L(\frac12)\) \(\approx\) \(2.799197947\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.43 + 3.42i)T \)
good7 \( 1 + (1.09 + 1.89i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.56 - 4.44i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.21 + 5.57i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.653 + 1.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.76 - 4.78i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.52 - 2.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.69 - 6.39i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.00 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.40 + 2.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.19 - 2.07i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 9.64T + 43T^{2} \)
47 \( 1 + (-1.71 - 2.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.09T + 53T^{2} \)
59 \( 1 + 5.99T + 59T^{2} \)
61 \( 1 + (-6.06 + 10.5i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-7.59 - 13.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.15 + 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.94 - 8.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.69 + 11.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 5.88T + 89T^{2} \)
97 \( 1 + (-7.09 + 12.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.328069439391671297235933508853, −7.78239012261943290581801143991, −7.01185879468813889435879520346, −6.32940865372221710524243067997, −5.53317942723806535473456283179, −4.51834588153203844404604830940, −3.74445125302212516687158822163, −3.07882204818339752375132563517, −1.90416351144252133116365199145, −1.01918327572288818694718984856, 0.942756794459645541894288790746, 2.14485223228297874144035102496, 2.82550201279448741728862516732, 3.88576271497291049322820145688, 4.42662766369728256979837754281, 5.72857775842007750439779068100, 6.36970529150785924930873560371, 6.65057417722436173682002865769, 7.938130170849205876272352386431, 8.668404358101265381412118439652

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