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Properties

Label 2-20-20.3-c19-0-11
Degree $2$
Conductor $20$
Sign $-0.803 - 0.594i$
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  + (−679. − 249. i)2-s + (1.17e4 − 1.17e4i)3-s + (3.99e5 + 3.39e5i)4-s + (−3.55e6 + 2.54e6i)5-s + (−1.09e7 + 5.07e6i)6-s + (1.37e8 + 1.37e8i)7-s + (−1.86e8 − 3.30e8i)8-s + 8.83e8i·9-s + (3.04e9 − 8.40e8i)10-s + 9.93e9i·11-s + (8.72e9 − 7.05e8i)12-s + (1.76e10 + 1.76e10i)13-s + (−5.90e10 − 1.27e11i)14-s + (−1.19e10 + 7.18e10i)15-s + (4.42e10 + 2.71e11i)16-s + (−5.52e11 + 5.52e11i)17-s + ⋯
L(s)  = 1  + (−0.938 − 0.345i)2-s + (0.346 − 0.346i)3-s + (0.761 + 0.647i)4-s + (−0.813 + 0.581i)5-s + (−0.444 + 0.205i)6-s + (1.28 + 1.28i)7-s + (−0.491 − 0.870i)8-s + 0.760i·9-s + (0.964 − 0.265i)10-s + 1.26i·11-s + (0.487 − 0.0394i)12-s + (0.462 + 0.462i)13-s + (−0.764 − 1.65i)14-s + (−0.0800 + 0.482i)15-s + (0.160 + 0.986i)16-s + (−1.12 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.594i)\, \overline{\Lambda}(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & (-0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.803 - 0.594i$
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: \((2,\ 20,\ (\ :19/2),\ -0.803 - 0.594i)\)

Particular Values

\(L(10)\) \(\approx\) \(1.057251560\)
\(L(\frac12)\) \(\approx\) \(1.057251560\)
\(L(\frac{21}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (679. + 249. i)T \)
5 \( 1 + (3.55e6 - 2.54e6i)T \)
good3 \( 1 + (-1.17e4 + 1.17e4i)T - 1.16e9iT^{2} \)
7 \( 1 + (-1.37e8 - 1.37e8i)T + 1.13e16iT^{2} \)
11 \( 1 - 9.93e9iT - 6.11e19T^{2} \)
13 \( 1 + (-1.76e10 - 1.76e10i)T + 1.46e21iT^{2} \)
17 \( 1 + (5.52e11 - 5.52e11i)T - 2.39e23iT^{2} \)
19 \( 1 + 8.68e11T + 1.97e24T^{2} \)
23 \( 1 + (-8.39e12 + 8.39e12i)T - 7.46e25iT^{2} \)
29 \( 1 + 4.44e13iT - 6.10e27T^{2} \)
31 \( 1 + 8.04e13iT - 2.16e28T^{2} \)
37 \( 1 + (1.08e14 - 1.08e14i)T - 6.24e29iT^{2} \)
41 \( 1 + 2.33e15T + 4.39e30T^{2} \)
43 \( 1 + (1.06e14 - 1.06e14i)T - 1.08e31iT^{2} \)
47 \( 1 + (1.54e15 + 1.54e15i)T + 5.88e31iT^{2} \)
53 \( 1 + (-4.02e15 - 4.02e15i)T + 5.77e32iT^{2} \)
59 \( 1 + 6.36e16T + 4.42e33T^{2} \)
61 \( 1 - 1.31e17T + 8.34e33T^{2} \)
67 \( 1 + (6.84e16 + 6.84e16i)T + 4.95e34iT^{2} \)
71 \( 1 - 1.35e17iT - 1.49e35T^{2} \)
73 \( 1 + (-3.63e17 - 3.63e17i)T + 2.53e35iT^{2} \)
79 \( 1 - 1.14e18T + 1.13e36T^{2} \)
83 \( 1 + (1.20e18 - 1.20e18i)T - 2.90e36iT^{2} \)
89 \( 1 + 6.30e18iT - 1.09e37T^{2} \)
97 \( 1 + (9.16e17 - 9.16e17i)T - 5.60e37iT^{2} \)
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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

−14.83767789229660160868067294781, −12.71995443836530105188090453777, −11.55456689619835659191984750072, −10.67317356677258111992693561616, −8.711098903880248146245608619356, −8.058391620655098201086253592164, −6.74671923426262724253034250853, −4.44375412553864968381448047207, −2.40666822584950569496279672165, −1.80611551851135304940071664032, 0.41436925211543639543078041040, 1.18275162570257281410218908755, 3.46332921300743891936575535641, 4.95795081817510402694432646861, 6.97609219572917210062747017175, 8.208550483222337683862054868911, 8.991086231055710615217336679345, 10.79939227456091900312221313393, 11.49069407170295385975876642339, 13.69569000624596609097477699009

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