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Properties

Label 2-20-20.3-c5-0-9
Degree $2$
Conductor $20$
Sign $0.897 + 0.441i$
Analytic cond. $3.20767$
Root an. cond. $1.79099$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.65 − 0.216i)2-s + (10.7 − 10.7i)3-s + (31.9 − 2.44i)4-s + (−55.8 + 1.12i)5-s + (58.4 − 63.1i)6-s + (26.5 + 26.5i)7-s + (179. − 20.6i)8-s + 11.5i·9-s + (−315. + 18.4i)10-s + 430. i·11-s + (316. − 369. i)12-s + (−557. − 557. i)13-s + (155. + 144. i)14-s + (−589. + 613. i)15-s + (1.01e3 − 155. i)16-s + (−780. + 780. i)17-s + ⋯
L(s)  = 1  + (0.999 − 0.0381i)2-s + (0.690 − 0.690i)3-s + (0.997 − 0.0763i)4-s + (−0.999 + 0.0201i)5-s + (0.663 − 0.716i)6-s + (0.204 + 0.204i)7-s + (0.993 − 0.114i)8-s + 0.0473i·9-s + (−0.998 + 0.0583i)10-s + 1.07i·11-s + (0.635 − 0.740i)12-s + (−0.915 − 0.915i)13-s + (0.212 + 0.197i)14-s + (−0.676 + 0.703i)15-s + (0.988 − 0.152i)16-s + (−0.655 + 0.655i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.897 + 0.441i$
Analytic conductor: \(3.20767\)
Root analytic conductor: \(1.79099\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5/2),\ 0.897 + 0.441i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.46575 - 0.573461i\)
\(L(\frac12)\) \(\approx\) \(2.46575 - 0.573461i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.65 + 0.216i)T \)
5 \( 1 + (55.8 - 1.12i)T \)
good3 \( 1 + (-10.7 + 10.7i)T - 243iT^{2} \)
7 \( 1 + (-26.5 - 26.5i)T + 1.68e4iT^{2} \)
11 \( 1 - 430. iT - 1.61e5T^{2} \)
13 \( 1 + (557. + 557. i)T + 3.71e5iT^{2} \)
17 \( 1 + (780. - 780. i)T - 1.41e6iT^{2} \)
19 \( 1 + 1.41e3T + 2.47e6T^{2} \)
23 \( 1 + (-3.11e3 + 3.11e3i)T - 6.43e6iT^{2} \)
29 \( 1 + 1.87e3iT - 2.05e7T^{2} \)
31 \( 1 + 5.92e3iT - 2.86e7T^{2} \)
37 \( 1 + (517. - 517. i)T - 6.93e7iT^{2} \)
41 \( 1 + 8.37e3T + 1.15e8T^{2} \)
43 \( 1 + (2.30e3 - 2.30e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (-1.70e4 - 1.70e4i)T + 2.29e8iT^{2} \)
53 \( 1 + (1.22e3 + 1.22e3i)T + 4.18e8iT^{2} \)
59 \( 1 - 2.34e3T + 7.14e8T^{2} \)
61 \( 1 + 2.85e4T + 8.44e8T^{2} \)
67 \( 1 + (2.46e4 + 2.46e4i)T + 1.35e9iT^{2} \)
71 \( 1 - 5.49e4iT - 1.80e9T^{2} \)
73 \( 1 + (-3.02e4 - 3.02e4i)T + 2.07e9iT^{2} \)
79 \( 1 + 3.81e4T + 3.07e9T^{2} \)
83 \( 1 + (-3.02e4 + 3.02e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 1.23e5iT - 5.58e9T^{2} \)
97 \( 1 + (-3.00e3 + 3.00e3i)T - 8.58e9iT^{2} \)
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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

−17.01748931855521604232010464442, −15.22374612766477619701288408471, −14.78589218367831381713508344143, −13.04292250123166286444526056481, −12.34613941757641022554963394236, −10.73760241814904292978099758010, −8.142908388312515044371724540468, −7.02420589680621387477570092118, −4.59206318510373821200587496895, −2.47173619153144202606170825036, 3.27286887027258335276004024275, 4.63411375687717856478201459695, 7.07176047350199867196525253060, 8.831781274715638506161999381963, 10.90106740540892538877384216735, 12.04860962531281215652676204325, 13.74436203128464800041641060160, 14.82937735948297343599916449856, 15.68377639129344098425469686952, 16.80904201828118728510471705546

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