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Properties

Label 2-4020-67.29-c1-0-28
Degree $2$
Conductor $4020$
Sign $0.766 + 0.641i$
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 + (1.22 + 2.11i)7-s + 9-s + (−1.39 − 2.40i)11-s + (0.5 − 0.866i)13-s − 15-s + (3.87 − 6.71i)17-s + (−3.64 + 6.30i)19-s + (1.22 + 2.11i)21-s + (0.334 − 0.579i)23-s + 25-s + 27-s + (−4.34 − 7.51i)29-s + (2.47 + 4.28i)31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + (0.461 + 0.798i)7-s + 0.333·9-s + (−0.419 − 0.726i)11-s + (0.138 − 0.240i)13-s − 0.258·15-s + (0.939 − 1.62i)17-s + (−0.835 + 1.44i)19-s + (0.266 + 0.461i)21-s + (0.0698 − 0.120i)23-s + 0.200·25-s + 0.192·27-s + (−0.805 − 1.39i)29-s + (0.443 + 0.768i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.766 + 0.641i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.766 + 0.641i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.130504525\)
\(L(\frac12)\) \(\approx\) \(2.130504525\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.58 - 7.35i)T \)
good7 \( 1 + (-1.22 - 2.11i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.39 + 2.40i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.87 + 6.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.64 - 6.30i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.334 + 0.579i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.34 + 7.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.47 - 4.28i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.86 + 8.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.30 + 3.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 4.28T + 43T^{2} \)
47 \( 1 + (-2.96 - 5.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 3.94T + 53T^{2} \)
59 \( 1 + 0.583T + 59T^{2} \)
61 \( 1 + (-0.949 + 1.64i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (1.74 + 3.02i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.24 + 3.88i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.89 + 5.02i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.02 + 6.97i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.596T + 89T^{2} \)
97 \( 1 + (-0.689 + 1.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.252353781158806606310132338448, −7.82949258964019606922246442620, −7.15109831598869145317321292997, −5.89998574199866512306791009873, −5.54821545076469787656909234205, −4.50232885095074626733778100881, −3.65685924145028758179533167521, −2.83270465143690942629223166734, −2.05187266393029350505778134849, −0.64699454257182999173659111662, 1.07282434126902277154638842149, 2.06817664326389638220960801811, 3.11793140738749782489634530253, 4.05136317577410669009516626853, 4.49873575349381049327831370599, 5.44117865640999772550608699838, 6.57387145956357863534419888806, 7.15459886271009283103043848865, 7.919784028955827482890754515895, 8.318030311053430617520201070598

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