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Properties

Label 2-20-20.3-c19-0-0
Degree $2$
Conductor $20$
Sign $-0.126 - 0.991i$
Analytic cond. $45.7633$
Root an. cond. $6.76486$
Motivic weight $19$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−190. − 698. i)2-s + (3.12e4 − 3.12e4i)3-s + (−4.51e5 + 2.65e5i)4-s + (6.48e5 + 4.31e6i)5-s + (−2.77e7 − 1.58e7i)6-s + (3.60e7 + 3.60e7i)7-s + (2.71e8 + 2.65e8i)8-s − 7.86e8i·9-s + (2.89e9 − 1.27e9i)10-s − 1.35e10i·11-s + (−5.80e9 + 2.24e10i)12-s + (−5.93e9 − 5.93e9i)13-s + (1.83e10 − 3.20e10i)14-s + (1.55e11 + 1.14e11i)15-s + (1.33e11 − 2.40e11i)16-s + (−6.05e11 + 6.05e11i)17-s + ⋯
L(s)  = 1  + (−0.262 − 0.964i)2-s + (0.915 − 0.915i)3-s + (−0.861 + 0.507i)4-s + (0.148 + 0.988i)5-s + (−1.12 − 0.642i)6-s + (0.337 + 0.337i)7-s + (0.715 + 0.698i)8-s − 0.676i·9-s + (0.915 − 0.403i)10-s − 1.73i·11-s + (−0.324 + 1.25i)12-s + (−0.155 − 0.155i)13-s + (0.236 − 0.413i)14-s + (1.04 + 0.769i)15-s + (0.485 − 0.874i)16-s + (−1.23 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.126 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.126 - 0.991i$
Analytic conductor: \(45.7633\)
Root analytic conductor: \(6.76486\)
Motivic weight: \(19\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :19/2),\ -0.126 - 0.991i)\)

Particular Values

\(L(10)\) \(\approx\) \(0.05835093413\)
\(L(\frac12)\) \(\approx\) \(0.05835093413\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (190. + 698. i)T \)
5 \( 1 + (-6.48e5 - 4.31e6i)T \)
good3 \( 1 + (-3.12e4 + 3.12e4i)T - 1.16e9iT^{2} \)
7 \( 1 + (-3.60e7 - 3.60e7i)T + 1.13e16iT^{2} \)
11 \( 1 + 1.35e10iT - 6.11e19T^{2} \)
13 \( 1 + (5.93e9 + 5.93e9i)T + 1.46e21iT^{2} \)
17 \( 1 + (6.05e11 - 6.05e11i)T - 2.39e23iT^{2} \)
19 \( 1 + 2.07e12T + 1.97e24T^{2} \)
23 \( 1 + (6.76e12 - 6.76e12i)T - 7.46e25iT^{2} \)
29 \( 1 + 1.06e14iT - 6.10e27T^{2} \)
31 \( 1 + 1.97e13iT - 2.16e28T^{2} \)
37 \( 1 + (-1.20e13 + 1.20e13i)T - 6.24e29iT^{2} \)
41 \( 1 + 2.08e15T + 4.39e30T^{2} \)
43 \( 1 + (1.49e15 - 1.49e15i)T - 1.08e31iT^{2} \)
47 \( 1 + (7.49e14 + 7.49e14i)T + 5.88e31iT^{2} \)
53 \( 1 + (-1.34e16 - 1.34e16i)T + 5.77e32iT^{2} \)
59 \( 1 + 4.82e16T + 4.42e33T^{2} \)
61 \( 1 + 7.43e16T + 8.34e33T^{2} \)
67 \( 1 + (-2.26e17 - 2.26e17i)T + 4.95e34iT^{2} \)
71 \( 1 + 2.81e17iT - 1.49e35T^{2} \)
73 \( 1 + (2.08e17 + 2.08e17i)T + 2.53e35iT^{2} \)
79 \( 1 + 1.57e18T + 1.13e36T^{2} \)
83 \( 1 + (-7.69e17 + 7.69e17i)T - 2.90e36iT^{2} \)
89 \( 1 - 4.16e18iT - 1.09e37T^{2} \)
97 \( 1 + (1.04e18 - 1.04e18i)T - 5.60e37iT^{2} \)
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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

−13.80833308243766034327632058814, −13.16185865821349220886474354769, −11.52592998052388654032908748050, −10.51071085574201841264292356993, −8.703157117085399682406159577158, −7.993465584998902359005577151677, −6.21014936940158216772665965630, −3.75333860423691554975963582179, −2.54617756245861611721410448741, −1.76145358196750232177353894169, 0.01412036807322498724565816121, 1.92758942981298588491817569705, 4.35066962918653064265392709038, 4.74361789619331451626877449206, 6.91760091454275445305161671951, 8.421702222618694270700252594104, 9.248736028186347623530802191109, 10.22969994373754065120835170675, 12.64670128080186023117658009960, 14.00164112985349379913490136720

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