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Properties

Label 2-20-20.3-c19-0-36
Degree $2$
Conductor $20$
Sign $0.960 + 0.278i$
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  + (77.0 + 719. i)2-s + (3.44e4 − 3.44e4i)3-s + (−5.12e5 + 1.10e5i)4-s + (−4.36e6 + 2.52e5i)5-s + (2.74e7 + 2.21e7i)6-s + (9.57e7 + 9.57e7i)7-s + (−1.19e8 − 3.60e8i)8-s − 1.20e9i·9-s + (−5.17e8 − 3.11e9i)10-s − 3.74e9i·11-s + (−1.38e10 + 2.14e10i)12-s + (−4.18e9 − 4.18e9i)13-s + (−6.15e10 + 7.62e10i)14-s + (−1.41e11 + 1.58e11i)15-s + (2.50e11 − 1.13e11i)16-s + (1.33e11 − 1.33e11i)17-s + ⋯
L(s)  = 1  + (0.106 + 0.994i)2-s + (1.00 − 1.00i)3-s + (−0.977 + 0.211i)4-s + (−0.998 + 0.0578i)5-s + (1.11 + 0.896i)6-s + (0.896 + 0.896i)7-s + (−0.314 − 0.949i)8-s − 1.03i·9-s + (−0.163 − 0.986i)10-s − 0.478i·11-s + (−0.772 + 1.19i)12-s + (−0.109 − 0.109i)13-s + (−0.796 + 0.986i)14-s + (−0.949 + 1.06i)15-s + (0.910 − 0.413i)16-s + (0.273 − 0.273i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.960 + 0.278i$
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: \((2,\ 20,\ (\ :19/2),\ 0.960 + 0.278i)\)

Particular Values

\(L(10)\) \(\approx\) \(2.320545439\)
\(L(\frac12)\) \(\approx\) \(2.320545439\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-77.0 - 719. i)T \)
5 \( 1 + (4.36e6 - 2.52e5i)T \)
good3 \( 1 + (-3.44e4 + 3.44e4i)T - 1.16e9iT^{2} \)
7 \( 1 + (-9.57e7 - 9.57e7i)T + 1.13e16iT^{2} \)
11 \( 1 + 3.74e9iT - 6.11e19T^{2} \)
13 \( 1 + (4.18e9 + 4.18e9i)T + 1.46e21iT^{2} \)
17 \( 1 + (-1.33e11 + 1.33e11i)T - 2.39e23iT^{2} \)
19 \( 1 + 2.29e11T + 1.97e24T^{2} \)
23 \( 1 + (6.38e12 - 6.38e12i)T - 7.46e25iT^{2} \)
29 \( 1 + 9.26e13iT - 6.10e27T^{2} \)
31 \( 1 + 1.84e14iT - 2.16e28T^{2} \)
37 \( 1 + (-7.69e14 + 7.69e14i)T - 6.24e29iT^{2} \)
41 \( 1 - 3.35e15T + 4.39e30T^{2} \)
43 \( 1 + (-4.55e15 + 4.55e15i)T - 1.08e31iT^{2} \)
47 \( 1 + (-2.45e15 - 2.45e15i)T + 5.88e31iT^{2} \)
53 \( 1 + (1.75e16 + 1.75e16i)T + 5.77e32iT^{2} \)
59 \( 1 - 1.27e17T + 4.42e33T^{2} \)
61 \( 1 + 2.11e16T + 8.34e33T^{2} \)
67 \( 1 + (1.04e17 + 1.04e17i)T + 4.95e34iT^{2} \)
71 \( 1 - 4.34e17iT - 1.49e35T^{2} \)
73 \( 1 + (5.76e17 + 5.76e17i)T + 2.53e35iT^{2} \)
79 \( 1 + 6.04e17T + 1.13e36T^{2} \)
83 \( 1 + (1.32e17 - 1.32e17i)T - 2.90e36iT^{2} \)
89 \( 1 - 1.33e18iT - 1.09e37T^{2} \)
97 \( 1 + (-6.21e18 + 6.21e18i)T - 5.60e37iT^{2} \)
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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

−14.13266251746798425791201945317, −12.85411038832162373936706883184, −11.66023079292174859083051890514, −9.063361700069365608840703575971, −8.020236715802457087439183224097, −7.52632829471097314735746030453, −5.78365527226295528687575308193, −4.03784286519260698646529824786, −2.44615387889664106732655102185, −0.64659255583768911407444522705, 1.13890292682385215837044935688, 2.84277377564929726467232733342, 4.09391810495449432119592164023, 4.60131909481511775411521540890, 7.78019915665876868599116361087, 8.795262116680210554583849731316, 10.17624671018917872046932808176, 11.09853336080922248267653328932, 12.53847084182976902411589977412, 14.26145945487742784545645472430

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