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Properties

Label 2-20-20.3-c9-0-15
Degree $2$
Conductor $20$
Sign $0.994 - 0.106i$
Analytic cond. $10.3007$
Root an. cond. $3.20947$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.87 + 21.5i)2-s + (−110. + 110. i)3-s + (−417. − 296. i)4-s + (404. − 1.33e3i)5-s + (−1.62e3 − 3.13e3i)6-s + (−1.21e3 − 1.21e3i)7-s + (9.25e3 − 6.96e3i)8-s − 4.67e3i·9-s + (2.60e4 + 1.79e4i)10-s + 3.90e4i·11-s + (7.87e4 − 1.33e4i)12-s + (−5.35e4 − 5.35e4i)13-s + (3.45e4 − 1.78e4i)14-s + (1.03e5 + 1.92e5i)15-s + (8.64e4 + 2.47e5i)16-s + (3.81e5 − 3.81e5i)17-s + ⋯
L(s)  = 1  + (−0.303 + 0.952i)2-s + (−0.786 + 0.786i)3-s + (−0.815 − 0.578i)4-s + (0.289 − 0.957i)5-s + (−0.510 − 0.988i)6-s + (−0.191 − 0.191i)7-s + (0.799 − 0.601i)8-s − 0.237i·9-s + (0.824 + 0.566i)10-s + 0.803i·11-s + (1.09 − 0.186i)12-s + (−0.519 − 0.519i)13-s + (0.240 − 0.124i)14-s + (0.525 + 0.980i)15-s + (0.329 + 0.944i)16-s + (1.10 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.994 - 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.994 - 0.106i$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :9/2),\ 0.994 - 0.106i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.877902 + 0.0467667i\)
\(L(\frac12)\) \(\approx\) \(0.877902 + 0.0467667i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.87 - 21.5i)T \)
5 \( 1 + (-404. + 1.33e3i)T \)
good3 \( 1 + (110. - 110. i)T - 1.96e4iT^{2} \)
7 \( 1 + (1.21e3 + 1.21e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 3.90e4iT - 2.35e9T^{2} \)
13 \( 1 + (5.35e4 + 5.35e4i)T + 1.06e10iT^{2} \)
17 \( 1 + (-3.81e5 + 3.81e5i)T - 1.18e11iT^{2} \)
19 \( 1 - 5.75e5T + 3.22e11T^{2} \)
23 \( 1 + (-1.06e6 + 1.06e6i)T - 1.80e12iT^{2} \)
29 \( 1 + 3.51e6iT - 1.45e13T^{2} \)
31 \( 1 - 4.45e6iT - 2.64e13T^{2} \)
37 \( 1 + (-5.44e6 + 5.44e6i)T - 1.29e14iT^{2} \)
41 \( 1 - 2.53e6T + 3.27e14T^{2} \)
43 \( 1 + (-2.19e7 + 2.19e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (1.86e7 + 1.86e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (-6.92e7 - 6.92e7i)T + 3.29e15iT^{2} \)
59 \( 1 + 1.64e8T + 8.66e15T^{2} \)
61 \( 1 + 1.57e6T + 1.16e16T^{2} \)
67 \( 1 + (2.14e8 + 2.14e8i)T + 2.72e16iT^{2} \)
71 \( 1 + 5.04e7iT - 4.58e16T^{2} \)
73 \( 1 + (-4.03e7 - 4.03e7i)T + 5.88e16iT^{2} \)
79 \( 1 - 3.84e8T + 1.19e17T^{2} \)
83 \( 1 + (5.30e8 - 5.30e8i)T - 1.86e17iT^{2} \)
89 \( 1 + 4.15e8iT - 3.50e17T^{2} \)
97 \( 1 + (-8.68e7 + 8.68e7i)T - 7.60e17iT^{2} \)
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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

−16.42518635648705049287275060495, −15.38373637214141179820634539064, −13.81224625567255673520622523076, −12.27448941794712496992356399524, −10.25377808537388010908018370175, −9.369699460243125733994373433445, −7.52520366971320603683811148547, −5.52924527468085664968882160620, −4.67959106870873589811047377285, −0.59319513444273192724249083278, 1.27200850515005002370190411841, 3.18766114246371566461576191394, 5.84206490551342533787220462462, 7.50143004250052909975370970099, 9.548773661382227053205207620188, 10.99067039324893847820198789394, 11.92728308099591186390531679945, 13.16281914384747282883433321091, 14.50523648536183465937700665647, 16.73982417587277666552769994397

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