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Properties

Label 2-4020-67.29-c1-0-5
Degree $2$
Conductor $4020$
Sign $-0.468 - 0.883i$
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 + (0.479 + 0.829i)7-s + 9-s + (2.72 + 4.71i)11-s + (−0.313 + 0.543i)13-s + 15-s + (0.164 − 0.285i)17-s + (−1.58 + 2.74i)19-s + (−0.479 − 0.829i)21-s + (2.10 − 3.64i)23-s + 25-s − 27-s + (3.47 + 6.02i)29-s + (−0.592 − 1.02i)31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + (0.181 + 0.313i)7-s + 0.333·9-s + (0.820 + 1.42i)11-s + (−0.0869 + 0.150i)13-s + 0.258·15-s + (0.0400 − 0.0693i)17-s + (−0.363 + 0.630i)19-s + (−0.104 − 0.181i)21-s + (0.439 − 0.760i)23-s + 0.200·25-s − 0.192·27-s + (0.645 + 1.11i)29-s + (−0.106 − 0.184i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.468 - 0.883i$
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: \((2,\ 4020,\ (\ :1/2),\ -0.468 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.103332421\)
\(L(\frac12)\) \(\approx\) \(1.103332421\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.549 - 8.16i)T \)
good7 \( 1 + (-0.479 - 0.829i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.72 - 4.71i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.313 - 0.543i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.164 + 0.285i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.58 - 2.74i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.10 + 3.64i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.47 - 6.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.592 + 1.02i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.869 - 1.50i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.61 - 7.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 5.91T + 43T^{2} \)
47 \( 1 + (5.13 + 8.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.34T + 53T^{2} \)
59 \( 1 + 5.10T + 59T^{2} \)
61 \( 1 + (3.31 - 5.74i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-1.82 - 3.16i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.21 - 5.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.94 + 12.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.70 + 9.87i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + (-1.57 + 2.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.721150953117678224161966354926, −7.88042030761450673409739324012, −7.07820542812720368320621632361, −6.60887521715437701181784858497, −5.75171888425743932044141348162, −4.70535517182107813691104709680, −4.42939457159299245852751286194, −3.34575261479573628815047536652, −2.15872814295123549575336829493, −1.20724838574477861133169644160, 0.39831985209363370379724209250, 1.31239077539756475802001909638, 2.76779419971518512976758314171, 3.68539005256597495827311362517, 4.36910121178514358167045705647, 5.23634971555380422982477301120, 6.11390946172192702702713052862, 6.56112652913665924930285717587, 7.57441480530126779314742712061, 8.047084444196797709807193211325

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