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Properties

Label 2-4020-67.37-c1-0-44
Degree $2$
Conductor $4020$
Sign $-0.556 + 0.830i$
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 + (−0.436 + 0.756i)7-s + 9-s + (1.09 − 1.89i)11-s + (−1.27 − 2.20i)13-s + 15-s + (−3.07 − 5.33i)17-s + (−1.66 − 2.89i)19-s + (−0.436 + 0.756i)21-s + (−2.10 − 3.64i)23-s + 25-s + 27-s + (−0.323 + 0.559i)29-s + (0.579 − 1.00i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (−0.165 + 0.285i)7-s + 0.333·9-s + (0.330 − 0.572i)11-s + (−0.353 − 0.611i)13-s + 0.258·15-s + (−0.746 − 1.29i)17-s + (−0.382 − 0.663i)19-s + (−0.0953 + 0.165i)21-s + (−0.438 − 0.759i)23-s + 0.200·25-s + 0.192·27-s + (−0.0600 + 0.103i)29-s + (0.104 − 0.180i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.556 + 0.830i$
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: \((2,\ 4020,\ (\ :1/2),\ -0.556 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.433557946\)
\(L(\frac12)\) \(\approx\) \(1.433557946\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.38 + 8.06i)T \)
good7 \( 1 + (0.436 - 0.756i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.09 + 1.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.27 + 2.20i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.07 + 5.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.66 + 2.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.10 + 3.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.323 - 0.559i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.579 + 1.00i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.36 + 4.09i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.00 - 6.93i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 9.15T + 43T^{2} \)
47 \( 1 + (6.21 - 10.7i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 8.66T + 53T^{2} \)
59 \( 1 + 3.64T + 59T^{2} \)
61 \( 1 + (2.99 + 5.18i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-5.31 + 9.20i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.0239 + 0.0414i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.57 - 2.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.29 + 3.97i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.07T + 89T^{2} \)
97 \( 1 + (-1.76 - 3.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.251975496014425298418104033476, −7.56131476896299887865495764987, −6.60472569713894426472938789987, −6.18609518299832888026411605458, −5.03033291700224657570173334032, −4.54404362350840407589444518373, −3.24963581614992514927976498737, −2.74726727943431886978661498159, −1.77066685599394709168343359423, −0.33692333150632903853650899980, 1.66674413900721078566571543655, 2.04240033240336090048071692791, 3.39881346689295320251949071929, 4.01732178892513985278401391628, 4.83905166782850015322815753269, 5.81107404189459967196921735580, 6.70189644700841956782056634604, 7.05910843354694195929504881107, 8.166246572575848603043488857438, 8.601339211930500387364972088890

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