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Properties

Label 2-4020-67.37-c1-0-34
Degree $2$
Conductor $4020$
Sign $0.710 + 0.703i$
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 + (0.440 − 0.763i)7-s + 9-s + (1.63 − 2.83i)11-s + (0.784 + 1.35i)13-s + 15-s + (−3.66 − 6.35i)17-s + (4.31 + 7.46i)19-s + (0.440 − 0.763i)21-s + (−1.92 − 3.32i)23-s + 25-s + 27-s + (−0.0681 + 0.118i)29-s + (−0.841 + 1.45i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (0.166 − 0.288i)7-s + 0.333·9-s + (0.493 − 0.854i)11-s + (0.217 + 0.376i)13-s + 0.258·15-s + (−0.889 − 1.54i)17-s + (0.989 + 1.71i)19-s + (0.0961 − 0.166i)21-s + (−0.400 − 0.694i)23-s + 0.200·25-s + 0.192·27-s + (−0.0126 + 0.0219i)29-s + (−0.151 + 0.261i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.710 + 0.703i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.766563654\)
\(L(\frac12)\) \(\approx\) \(2.766563654\)
\(L(\frac{3}{2})\) 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 \( 1 + (7.65 + 2.88i)T \)
good7 \( 1 + (-0.440 + 0.763i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.63 + 2.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.784 - 1.35i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.66 + 6.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.31 - 7.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.92 + 3.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0681 - 0.118i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.841 - 1.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.66 + 2.88i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.87 + 4.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 + (-5.00 + 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 1.34T + 53T^{2} \)
59 \( 1 - 2.31T + 59T^{2} \)
61 \( 1 + (2.41 + 4.18i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (4.49 - 7.79i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.99 + 6.91i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.710 - 1.23i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.575 + 0.996i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.60T + 89T^{2} \)
97 \( 1 + (-1.95 - 3.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.486538052113449155642139525546, −7.55777286675039724048070682732, −7.05009283372167345433910995665, −6.08520448863893821805312718038, −5.49304461424655763585628509841, −4.40632952300449304896255815018, −3.73752139481972614773868959659, −2.81980426048259120065089011873, −1.90045077095230565948785585028, −0.797481871910535157135068308679, 1.21845514504784185070885805422, 2.15021850169171362069797114235, 2.92113752919610766061009149446, 4.02179830847671404936434860152, 4.63468971378026397698735314248, 5.63885702022846367737118194897, 6.32778059854314770289060333539, 7.19093203640014938349621202325, 7.74987587285272790350091126373, 8.743391763513480416899556393973

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