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Properties

Label 2-20-20.19-c6-0-10
Degree $2$
Conductor $20$
Sign $-0.831 + 0.555i$
Analytic cond. $4.60108$
Root an. cond. $2.14501$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.42 − 2.98i)2-s + 6.08·3-s + (46.1 + 44.3i)4-s + (26.7 − 122. i)5-s + (−45.1 − 18.1i)6-s − 422.·7-s + (−209. − 466. i)8-s − 691.·9-s + (−563. + 826. i)10-s − 1.74e3i·11-s + (280. + 269. i)12-s − 898. i·13-s + (3.13e3 + 1.26e3i)14-s + (163. − 743. i)15-s + (163. + 4.09e3i)16-s + 5.54e3i·17-s + ⋯
L(s)  = 1  + (−0.927 − 0.373i)2-s + 0.225·3-s + (0.721 + 0.692i)4-s + (0.214 − 0.976i)5-s + (−0.209 − 0.0842i)6-s − 1.23·7-s + (−0.410 − 0.912i)8-s − 0.949·9-s + (−0.563 + 0.826i)10-s − 1.31i·11-s + (0.162 + 0.156i)12-s − 0.409i·13-s + (1.14 + 0.459i)14-s + (0.0483 − 0.220i)15-s + (0.0398 + 0.999i)16-s + 1.12i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.831 + 0.555i$
Analytic conductor: \(4.60108\)
Root analytic conductor: \(2.14501\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :3),\ -0.831 + 0.555i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.175718 - 0.579029i\)
\(L(\frac12)\) \(\approx\) \(0.175718 - 0.579029i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (7.42 + 2.98i)T \)
5 \( 1 + (-26.7 + 122. i)T \)
good3 \( 1 - 6.08T + 729T^{2} \)
7 \( 1 + 422.T + 1.17e5T^{2} \)
11 \( 1 + 1.74e3iT - 1.77e6T^{2} \)
13 \( 1 + 898. iT - 4.82e6T^{2} \)
17 \( 1 - 5.54e3iT - 2.41e7T^{2} \)
19 \( 1 + 8.90e3iT - 4.70e7T^{2} \)
23 \( 1 - 3.74e3T + 1.48e8T^{2} \)
29 \( 1 - 3.11e4T + 5.94e8T^{2} \)
31 \( 1 - 2.26e4iT - 8.87e8T^{2} \)
37 \( 1 + 1.49e4iT - 2.56e9T^{2} \)
41 \( 1 - 6.59e3T + 4.75e9T^{2} \)
43 \( 1 - 8.10e4T + 6.32e9T^{2} \)
47 \( 1 - 3.02e4T + 1.07e10T^{2} \)
53 \( 1 + 1.72e4iT - 2.21e10T^{2} \)
59 \( 1 + 2.94e5iT - 4.21e10T^{2} \)
61 \( 1 + 2.11e5T + 5.15e10T^{2} \)
67 \( 1 + 4.52e5T + 9.04e10T^{2} \)
71 \( 1 + 2.46e5iT - 1.28e11T^{2} \)
73 \( 1 + 4.76e5iT - 1.51e11T^{2} \)
79 \( 1 - 6.80e5iT - 2.43e11T^{2} \)
83 \( 1 - 1.03e6T + 3.26e11T^{2} \)
89 \( 1 + 9.37e5T + 4.96e11T^{2} \)
97 \( 1 + 9.57e5iT - 8.32e11T^{2} \)
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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

−16.65917089916769397377608572310, −15.72153213233864935822863720054, −13.51075972196956083413546049829, −12.39392215268079076417852694500, −10.81035947462993591145770490198, −9.217265049576835235030823159735, −8.384353571501310968280877112117, −6.15433402694385642255788833689, −3.08838559297919779217472191904, −0.48370692648013842591907972715, 2.66674108907593586788325619207, 6.17226977573542349714813976423, 7.39505100162711112407540353313, 9.323791301029476132710498528966, 10.27212238143467060006635600067, 11.90991675795377289326230580465, 14.01416921586475096717745060100, 15.08725121388942647016341936360, 16.34784754556608024164624324927, 17.59610747657653431858002194558

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