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Properties

Label 2-20-5.4-c11-0-5
Degree $2$
Conductor $20$
Sign $-0.807 - 0.589i$
Analytic cond. $15.3668$
Root an. cond. $3.92005$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 763. i·3-s + (−5.64e3 − 4.11e3i)5-s − 5.96e4i·7-s − 4.05e5·9-s + 9.34e5·11-s + 1.09e6i·13-s + (−3.14e6 + 4.30e6i)15-s − 1.51e6i·17-s − 5.92e6·19-s − 4.55e7·21-s − 1.03e7i·23-s + (1.49e7 + 4.64e7i)25-s + 1.74e8i·27-s − 1.12e8·29-s + 1.59e8·31-s + ⋯
L(s)  = 1  − 1.81i·3-s + (−0.807 − 0.589i)5-s − 1.34i·7-s − 2.28·9-s + 1.74·11-s + 0.818i·13-s + (−1.06 + 1.46i)15-s − 0.259i·17-s − 0.548·19-s − 2.43·21-s − 0.335i·23-s + (0.305 + 0.952i)25-s + 2.33i·27-s − 1.01·29-s + 0.997·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.807 - 0.589i$
Analytic conductor: \(15.3668\)
Root analytic conductor: \(3.92005\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :11/2),\ -0.807 - 0.589i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.362302 + 1.11162i\)
\(L(\frac12)\) \(\approx\) \(0.362302 + 1.11162i\)
\(L(\frac{13}{2})\) 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 \( 1 + (5.64e3 + 4.11e3i)T \)
good3 \( 1 + 763. iT - 1.77e5T^{2} \)
7 \( 1 + 5.96e4iT - 1.97e9T^{2} \)
11 \( 1 - 9.34e5T + 2.85e11T^{2} \)
13 \( 1 - 1.09e6iT - 1.79e12T^{2} \)
17 \( 1 + 1.51e6iT - 3.42e13T^{2} \)
19 \( 1 + 5.92e6T + 1.16e14T^{2} \)
23 \( 1 + 1.03e7iT - 9.52e14T^{2} \)
29 \( 1 + 1.12e8T + 1.22e16T^{2} \)
31 \( 1 - 1.59e8T + 2.54e16T^{2} \)
37 \( 1 + 1.08e8iT - 1.77e17T^{2} \)
41 \( 1 - 2.91e8T + 5.50e17T^{2} \)
43 \( 1 + 1.33e9iT - 9.29e17T^{2} \)
47 \( 1 + 3.70e8iT - 2.47e18T^{2} \)
53 \( 1 + 7.84e8iT - 9.26e18T^{2} \)
59 \( 1 + 5.39e9T + 3.01e19T^{2} \)
61 \( 1 - 4.93e9T + 4.35e19T^{2} \)
67 \( 1 + 4.84e9iT - 1.22e20T^{2} \)
71 \( 1 + 1.78e10T + 2.31e20T^{2} \)
73 \( 1 - 1.05e9iT - 3.13e20T^{2} \)
79 \( 1 + 2.31e10T + 7.47e20T^{2} \)
83 \( 1 - 7.06e9iT - 1.28e21T^{2} \)
89 \( 1 - 2.31e10T + 2.77e21T^{2} \)
97 \( 1 - 1.70e10iT - 7.15e21T^{2} \)
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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

−14.42437159548384731841661058623, −13.48764836735352216866190151112, −12.25034420721231202784924341150, −11.37804998786094487461636830001, −8.829821244835246597735588814098, −7.44293996862300070588184708152, −6.59261101756165822847699551054, −4.01883153866605633390172180131, −1.51405254075637202225815610277, −0.49855675524588937479191597739, 3.08076055966809205735905963620, 4.29483003003891031541488866883, 5.98277043353646987901986350020, 8.516410159212884666876536048519, 9.598125691145073910883086795728, 11.04829271388876370589799825656, 11.98326293103620095250885221949, 14.66800996705269890330610115835, 15.10476386838541143206354502550, 16.09638018990047739817099514389

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