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Properties

Label 2-20-20.3-c7-0-9
Degree $2$
Conductor $20$
Sign $-0.533 + 0.846i$
Analytic cond. $6.24770$
Root an. cond. $2.49954$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.1 − 1.94i)2-s + (−31.8 + 31.8i)3-s + (120. + 43.4i)4-s + (92.5 + 263. i)5-s + (417. − 292. i)6-s + (−789. − 789. i)7-s + (−1.25e3 − 719. i)8-s + 156. i·9-s + (−517. − 3.11e3i)10-s − 4.79e3i·11-s + (−5.22e3 + 2.45e3i)12-s + (−5.75e3 − 5.75e3i)13-s + (7.25e3 + 1.03e4i)14-s + (−1.13e4 − 5.45e3i)15-s + (1.26e4 + 1.04e4i)16-s + (1.70e4 − 1.70e4i)17-s + ⋯
L(s)  = 1  + (−0.985 − 0.172i)2-s + (−0.681 + 0.681i)3-s + (0.940 + 0.339i)4-s + (0.331 + 0.943i)5-s + (0.788 − 0.553i)6-s + (−0.869 − 0.869i)7-s + (−0.868 − 0.496i)8-s + 0.0714i·9-s + (−0.163 − 0.986i)10-s − 1.08i·11-s + (−0.872 + 0.409i)12-s + (−0.726 − 0.726i)13-s + (0.706 + 1.00i)14-s + (−0.868 − 0.417i)15-s + (0.769 + 0.638i)16-s + (0.842 − 0.842i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.846i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.533 + 0.846i$
Analytic conductor: \(6.24770\)
Root analytic conductor: \(2.49954\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :7/2),\ -0.533 + 0.846i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0890626 - 0.161398i\)
\(L(\frac12)\) \(\approx\) \(0.0890626 - 0.161398i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.1 + 1.94i)T \)
5 \( 1 + (-92.5 - 263. i)T \)
good3 \( 1 + (31.8 - 31.8i)T - 2.18e3iT^{2} \)
7 \( 1 + (789. + 789. i)T + 8.23e5iT^{2} \)
11 \( 1 + 4.79e3iT - 1.94e7T^{2} \)
13 \( 1 + (5.75e3 + 5.75e3i)T + 6.27e7iT^{2} \)
17 \( 1 + (-1.70e4 + 1.70e4i)T - 4.10e8iT^{2} \)
19 \( 1 + 8.19e3T + 8.93e8T^{2} \)
23 \( 1 + (2.91e4 - 2.91e4i)T - 3.40e9iT^{2} \)
29 \( 1 + 2.36e5iT - 1.72e10T^{2} \)
31 \( 1 - 9.70e4iT - 2.75e10T^{2} \)
37 \( 1 + (2.09e5 - 2.09e5i)T - 9.49e10iT^{2} \)
41 \( 1 + 2.65e5T + 1.94e11T^{2} \)
43 \( 1 + (2.98e5 - 2.98e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (-3.54e5 - 3.54e5i)T + 5.06e11iT^{2} \)
53 \( 1 + (-4.23e5 - 4.23e5i)T + 1.17e12iT^{2} \)
59 \( 1 + 7.35e5T + 2.48e12T^{2} \)
61 \( 1 + 6.74e5T + 3.14e12T^{2} \)
67 \( 1 + (3.12e6 + 3.12e6i)T + 6.06e12iT^{2} \)
71 \( 1 - 5.87e5iT - 9.09e12T^{2} \)
73 \( 1 + (4.14e5 + 4.14e5i)T + 1.10e13iT^{2} \)
79 \( 1 + 8.15e6T + 1.92e13T^{2} \)
83 \( 1 + (-2.94e6 + 2.94e6i)T - 2.71e13iT^{2} \)
89 \( 1 + 6.58e6iT - 4.42e13T^{2} \)
97 \( 1 + (1.32e6 - 1.32e6i)T - 8.07e13iT^{2} \)
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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

−16.57704255481664413711335899213, −15.51458783966698891216834264696, −13.68364009523807335234780923732, −11.63728761350104300323397456470, −10.42090013185402165880987784066, −9.877486007031403714226639838620, −7.57870880093710006178169548468, −6.03432158433410189716024322912, −3.15780807240926188306674589447, −0.14853330515992410077279842181, 1.76085432988807134730026598172, 5.68794954041938481616763079699, 6.97882542662502709455204259593, 8.868490933497936025047035589353, 9.971868394056412018445672081376, 12.16158068809581384520036627760, 12.49083127404836717850697697369, 14.93533773364445549921920867775, 16.37587553743819877120801888637, 17.20893104874371388312380665994

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