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Properties

Label 2-20-5.2-c10-0-2
Degree $2$
Conductor $20$
Sign $0.969 - 0.243i$
Analytic cond. $12.7071$
Root an. cond. $3.56470$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (217. − 217. i)3-s + (−944. + 2.97e3i)5-s + (2.15e4 + 2.15e4i)7-s − 3.57e4i·9-s − 8.59e4·11-s + (4.15e5 − 4.15e5i)13-s + (4.42e5 + 8.54e5i)15-s + (1.81e6 + 1.81e6i)17-s − 1.42e6i·19-s + 9.37e6·21-s + (−2.90e6 + 2.90e6i)23-s + (−7.97e6 − 5.62e6i)25-s + (5.06e6 + 5.06e6i)27-s − 1.33e7i·29-s + 4.49e6·31-s + ⋯
L(s)  = 1  + (0.896 − 0.896i)3-s + (−0.302 + 0.953i)5-s + (1.28 + 1.28i)7-s − 0.605i·9-s − 0.533·11-s + (1.11 − 1.11i)13-s + (0.583 + 1.12i)15-s + (1.27 + 1.27i)17-s − 0.574i·19-s + 2.29·21-s + (−0.450 + 0.450i)23-s + (−0.817 − 0.576i)25-s + (0.353 + 0.353i)27-s − 0.650i·29-s + 0.157·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.969 - 0.243i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ 0.969 - 0.243i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.57371 + 0.318651i\)
\(L(\frac12)\) \(\approx\) \(2.57371 + 0.318651i\)
\(L(6)\) 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 \( 1 + (944. - 2.97e3i)T \)
good3 \( 1 + (-217. + 217. i)T - 5.90e4iT^{2} \)
7 \( 1 + (-2.15e4 - 2.15e4i)T + 2.82e8iT^{2} \)
11 \( 1 + 8.59e4T + 2.59e10T^{2} \)
13 \( 1 + (-4.15e5 + 4.15e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (-1.81e6 - 1.81e6i)T + 2.01e12iT^{2} \)
19 \( 1 + 1.42e6iT - 6.13e12T^{2} \)
23 \( 1 + (2.90e6 - 2.90e6i)T - 4.14e13iT^{2} \)
29 \( 1 + 1.33e7iT - 4.20e14T^{2} \)
31 \( 1 - 4.49e6T + 8.19e14T^{2} \)
37 \( 1 + (5.25e6 + 5.25e6i)T + 4.80e15iT^{2} \)
41 \( 1 + 1.21e8T + 1.34e16T^{2} \)
43 \( 1 + (-1.73e7 + 1.73e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (2.68e7 + 2.68e7i)T + 5.25e16iT^{2} \)
53 \( 1 + (-1.97e8 + 1.97e8i)T - 1.74e17iT^{2} \)
59 \( 1 - 2.66e8iT - 5.11e17T^{2} \)
61 \( 1 + 6.61e8T + 7.13e17T^{2} \)
67 \( 1 + (1.44e9 + 1.44e9i)T + 1.82e18iT^{2} \)
71 \( 1 + 1.11e9T + 3.25e18T^{2} \)
73 \( 1 + (-1.83e7 + 1.83e7i)T - 4.29e18iT^{2} \)
79 \( 1 + 2.56e9iT - 9.46e18T^{2} \)
83 \( 1 + (-8.98e8 + 8.98e8i)T - 1.55e19iT^{2} \)
89 \( 1 - 4.22e9iT - 3.11e19T^{2} \)
97 \( 1 + (-5.59e9 - 5.59e9i)T + 7.37e19iT^{2} \)
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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

−15.40665612837333459053973837873, −14.74619022366453558051623504582, −13.48921645432792620834683486436, −12.06914889622395354997353112068, −10.67613082717406106991173181865, −8.374020853669571629173369160748, −7.81008710515606301820042822748, −5.78651678765939391757999044341, −3.08391020231401828248508542159, −1.74890075358159329842293616938, 1.22381773318094462042799127297, 3.78376848499172644715850178804, 4.82400005630219287399421328006, 7.73403214956638393789574411388, 8.798964058273464700412542132826, 10.23811630297422451660867704604, 11.70984057360352908706670138012, 13.70611821006669612638877177797, 14.42459866605060854535044848176, 15.99017306284712337407452366417

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