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Properties

Label 2-20-20.3-c9-0-10
Degree $2$
Conductor $20$
Sign $0.439 - 0.898i$
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  + (20.7 − 9.06i)2-s + (−67.5 + 67.5i)3-s + (347. − 375. i)4-s + (−928. + 1.04e3i)5-s + (−787. + 2.01e3i)6-s + (5.70e3 + 5.70e3i)7-s + (3.79e3 − 1.09e4i)8-s + 1.05e4i·9-s + (−9.76e3 + 3.00e4i)10-s + 5.14e4i·11-s + (1.91e3 + 4.88e4i)12-s + (2.48e3 + 2.48e3i)13-s + (1.70e5 + 6.65e4i)14-s + (−7.88e3 − 1.33e5i)15-s + (−2.05e4 − 2.61e5i)16-s + (−9.14e4 + 9.14e4i)17-s + ⋯
L(s)  = 1  + (0.916 − 0.400i)2-s + (−0.481 + 0.481i)3-s + (0.678 − 0.734i)4-s + (−0.664 + 0.747i)5-s + (−0.248 + 0.634i)6-s + (0.898 + 0.898i)7-s + (0.327 − 0.944i)8-s + 0.536i·9-s + (−0.308 + 0.951i)10-s + 1.05i·11-s + (0.0267 + 0.680i)12-s + (0.0241 + 0.0241i)13-s + (1.18 + 0.462i)14-s + (−0.0401 − 0.679i)15-s + (−0.0783 − 0.996i)16-s + (−0.265 + 0.265i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.439 - 0.898i$
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: \((2,\ 20,\ (\ :9/2),\ 0.439 - 0.898i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.96859 + 1.22807i\)
\(L(\frac12)\) \(\approx\) \(1.96859 + 1.22807i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-20.7 + 9.06i)T \)
5 \( 1 + (928. - 1.04e3i)T \)
good3 \( 1 + (67.5 - 67.5i)T - 1.96e4iT^{2} \)
7 \( 1 + (-5.70e3 - 5.70e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 5.14e4iT - 2.35e9T^{2} \)
13 \( 1 + (-2.48e3 - 2.48e3i)T + 1.06e10iT^{2} \)
17 \( 1 + (9.14e4 - 9.14e4i)T - 1.18e11iT^{2} \)
19 \( 1 - 6.84e5T + 3.22e11T^{2} \)
23 \( 1 + (1.13e6 - 1.13e6i)T - 1.80e12iT^{2} \)
29 \( 1 + 6.65e6iT - 1.45e13T^{2} \)
31 \( 1 - 3.48e6iT - 2.64e13T^{2} \)
37 \( 1 + (-3.81e6 + 3.81e6i)T - 1.29e14iT^{2} \)
41 \( 1 + 1.31e7T + 3.27e14T^{2} \)
43 \( 1 + (-1.27e7 + 1.27e7i)T - 5.02e14iT^{2} \)
47 \( 1 + (-3.44e7 - 3.44e7i)T + 1.11e15iT^{2} \)
53 \( 1 + (-6.05e7 - 6.05e7i)T + 3.29e15iT^{2} \)
59 \( 1 - 6.68e7T + 8.66e15T^{2} \)
61 \( 1 - 1.28e8T + 1.16e16T^{2} \)
67 \( 1 + (-4.68e7 - 4.68e7i)T + 2.72e16iT^{2} \)
71 \( 1 + 2.72e8iT - 4.58e16T^{2} \)
73 \( 1 + (-1.23e8 - 1.23e8i)T + 5.88e16iT^{2} \)
79 \( 1 + 4.47e8T + 1.19e17T^{2} \)
83 \( 1 + (-2.50e8 + 2.50e8i)T - 1.86e17iT^{2} \)
89 \( 1 + 6.78e8iT - 3.50e17T^{2} \)
97 \( 1 + (3.98e8 - 3.98e8i)T - 7.60e17iT^{2} \)
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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

−15.80725133805666343909734862089, −15.19277643861693015152803128236, −13.94195367594067199499203972632, −12.03884510718589403903513080264, −11.35501975643801866833824951663, −10.06982327994483221794145719596, −7.55724876345896745731101651161, −5.59462198193081718890748692840, −4.26454737745702485617154541860, −2.23179778216216886083470072214, 0.911715151513941445823501899774, 3.82571469029910060218471189786, 5.34587527173397157559650823932, 7.07583207817681003331329270161, 8.354721672257577663536928317389, 11.18600396199309263094942273090, 12.02756559883319239389269501807, 13.33535960250162388568692349002, 14.50234671682688788646653113186, 16.05445730554187238810464402056

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