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Properties

Label 2-20-20.3-c19-0-41
Degree $2$
Conductor $20$
Sign $-0.985 + 0.169i$
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  + (−7.75 − 724. i)2-s + (−1.23e4 + 1.23e4i)3-s + (−5.24e5 + 1.12e4i)4-s + (3.33e6 − 2.82e6i)5-s + (9.06e6 + 8.86e6i)6-s + (−3.93e7 − 3.93e7i)7-s + (1.21e7 + 3.79e8i)8-s + 8.55e8i·9-s + (−2.06e9 − 2.39e9i)10-s − 1.11e10i·11-s + (6.35e9 − 6.62e9i)12-s + (2.84e10 + 2.84e10i)13-s + (−2.81e10 + 2.87e10i)14-s + (−6.33e9 + 7.62e10i)15-s + (2.74e11 − 1.17e10i)16-s + (1.64e11 − 1.64e11i)17-s + ⋯
L(s)  = 1  + (−0.0107 − 0.999i)2-s + (−0.363 + 0.363i)3-s + (−0.999 + 0.0214i)4-s + (0.763 − 0.646i)5-s + (0.367 + 0.359i)6-s + (−0.368 − 0.368i)7-s + (0.0321 + 0.999i)8-s + 0.736i·9-s + (−0.654 − 0.756i)10-s − 1.43i·11-s + (0.355 − 0.370i)12-s + (0.743 + 0.743i)13-s + (−0.364 + 0.372i)14-s + (−0.0425 + 0.511i)15-s + (0.999 − 0.0428i)16-s + (0.336 − 0.336i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.985 + 0.169i$
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: \((2,\ 20,\ (\ :19/2),\ -0.985 + 0.169i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.279291272\)
\(L(\frac12)\) \(\approx\) \(1.279291272\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.75 + 724. i)T \)
5 \( 1 + (-3.33e6 + 2.82e6i)T \)
good3 \( 1 + (1.23e4 - 1.23e4i)T - 1.16e9iT^{2} \)
7 \( 1 + (3.93e7 + 3.93e7i)T + 1.13e16iT^{2} \)
11 \( 1 + 1.11e10iT - 6.11e19T^{2} \)
13 \( 1 + (-2.84e10 - 2.84e10i)T + 1.46e21iT^{2} \)
17 \( 1 + (-1.64e11 + 1.64e11i)T - 2.39e23iT^{2} \)
19 \( 1 - 1.17e12T + 1.97e24T^{2} \)
23 \( 1 + (-1.75e12 + 1.75e12i)T - 7.46e25iT^{2} \)
29 \( 1 + 7.15e13iT - 6.10e27T^{2} \)
31 \( 1 - 4.06e13iT - 2.16e28T^{2} \)
37 \( 1 + (1.29e14 - 1.29e14i)T - 6.24e29iT^{2} \)
41 \( 1 + 2.98e15T + 4.39e30T^{2} \)
43 \( 1 + (1.92e15 - 1.92e15i)T - 1.08e31iT^{2} \)
47 \( 1 + (-2.24e15 - 2.24e15i)T + 5.88e31iT^{2} \)
53 \( 1 + (6.19e15 + 6.19e15i)T + 5.77e32iT^{2} \)
59 \( 1 + 1.17e17T + 4.42e33T^{2} \)
61 \( 1 - 6.09e16T + 8.34e33T^{2} \)
67 \( 1 + (2.46e17 + 2.46e17i)T + 4.95e34iT^{2} \)
71 \( 1 - 2.33e17iT - 1.49e35T^{2} \)
73 \( 1 + (5.98e17 + 5.98e17i)T + 2.53e35iT^{2} \)
79 \( 1 - 1.76e18T + 1.13e36T^{2} \)
83 \( 1 + (-1.13e17 + 1.13e17i)T - 2.90e36iT^{2} \)
89 \( 1 - 9.66e16iT - 1.09e37T^{2} \)
97 \( 1 + (-6.25e18 + 6.25e18i)T - 5.60e37iT^{2} \)
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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

−13.48421682026811284166530879304, −11.77251712206214744547218563126, −10.68951119067380567528206942702, −9.574463431188930024030788208097, −8.374969435379586421198705741095, −5.89851397997843639421396494766, −4.73883923942093473801449739538, −3.24628986778328832995646208558, −1.59390998074522306370722773596, −0.40312712503602738943424112214, 1.33031973997975857320525150871, 3.37282271372355988392590415318, 5.35403757189816035040820007546, 6.37283527392306420043931156863, 7.34940325258482787584830850997, 9.165736326414539297656473807394, 10.21947224520104213244605322432, 12.29116902801280052567103452747, 13.33478712096869937472896993559, 14.73204276292339366824076748484

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