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Properties

Label 2-20-4.3-c10-0-7
Degree $2$
Conductor $20$
Sign $-0.820 - 0.571i$
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  + (30.5 + 9.58i)2-s + 321. i·3-s + (840. + 585. i)4-s − 1.39e3·5-s + (−3.08e3 + 9.82e3i)6-s + 9.88e3i·7-s + (2.00e4 + 2.59e4i)8-s − 4.46e4·9-s + (−4.26e4 − 1.33e4i)10-s − 1.16e5i·11-s + (−1.88e5 + 2.70e5i)12-s − 5.22e5·13-s + (−9.47e4 + 3.01e5i)14-s − 4.49e5i·15-s + (3.63e5 + 9.83e5i)16-s + 1.21e6·17-s + ⋯
L(s)  = 1  + (0.954 + 0.299i)2-s + 1.32i·3-s + (0.820 + 0.571i)4-s − 0.447·5-s + (−0.397 + 1.26i)6-s + 0.587i·7-s + (0.611 + 0.791i)8-s − 0.755·9-s + (−0.426 − 0.133i)10-s − 0.722i·11-s + (−0.757 + 1.08i)12-s − 1.40·13-s + (−0.176 + 0.560i)14-s − 0.592i·15-s + (0.346 + 0.938i)16-s + 0.857·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.820 - 0.571i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ -0.820 - 0.571i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.822496 + 2.61893i\)
\(L(\frac12)\) \(\approx\) \(0.822496 + 2.61893i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-30.5 - 9.58i)T \)
5 \( 1 + 1.39e3T \)
good3 \( 1 - 321. iT - 5.90e4T^{2} \)
7 \( 1 - 9.88e3iT - 2.82e8T^{2} \)
11 \( 1 + 1.16e5iT - 2.59e10T^{2} \)
13 \( 1 + 5.22e5T + 1.37e11T^{2} \)
17 \( 1 - 1.21e6T + 2.01e12T^{2} \)
19 \( 1 - 1.43e6iT - 6.13e12T^{2} \)
23 \( 1 - 6.87e6iT - 4.14e13T^{2} \)
29 \( 1 - 2.86e7T + 4.20e14T^{2} \)
31 \( 1 + 3.98e7iT - 8.19e14T^{2} \)
37 \( 1 - 1.09e8T + 4.80e15T^{2} \)
41 \( 1 - 3.34e7T + 1.34e16T^{2} \)
43 \( 1 + 7.92e7iT - 2.16e16T^{2} \)
47 \( 1 - 4.50e8iT - 5.25e16T^{2} \)
53 \( 1 + 2.29e8T + 1.74e17T^{2} \)
59 \( 1 + 1.31e9iT - 5.11e17T^{2} \)
61 \( 1 + 8.17e8T + 7.13e17T^{2} \)
67 \( 1 - 5.25e8iT - 1.82e18T^{2} \)
71 \( 1 - 1.13e9iT - 3.25e18T^{2} \)
73 \( 1 - 1.23e9T + 4.29e18T^{2} \)
79 \( 1 - 1.72e9iT - 9.46e18T^{2} \)
83 \( 1 + 5.52e9iT - 1.55e19T^{2} \)
89 \( 1 + 3.77e8T + 3.11e19T^{2} \)
97 \( 1 + 3.29e9T + 7.37e19T^{2} \)
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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

−16.07352299274851534649842385942, −15.19034474887845622781231577743, −14.26834441142729827248160131399, −12.41604402627618055891120680708, −11.25692353494717468652734188835, −9.710192765618704697602544298942, −7.84262182341129205932361273067, −5.69042006392648046807862366879, −4.41457567644187698317065352286, −3.00849933735923930140084693333, 0.900141003377172102243341960963, 2.56141737486287790049341485353, 4.67004638485823272368034177106, 6.72613592142457110225241296676, 7.59624332067917486933577867642, 10.26511361488160166925858564693, 12.01895956989654157296132536026, 12.59035468875523074091134379623, 13.89185737040488758819799791300, 14.95560918790012228308385073381

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