Properties

Label 2-20-20.19-c2-0-3
Degree $2$
Conductor $20$
Sign $0.599 + 0.799i$
Analytic cond. $0.544960$
Root an. cond. $0.738214$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 4·4-s + (3 + 4i)5-s + 8i·8-s − 9·9-s + (8 − 6i)10-s − 24i·13-s + 16·16-s + 16i·17-s + 18i·18-s + (−12 − 16i)20-s + (−7 + 24i)25-s − 48·26-s + 42·29-s − 32i·32-s + ⋯
L(s)  = 1  i·2-s − 4-s + (0.600 + 0.800i)5-s + i·8-s − 9-s + (0.800 − 0.600i)10-s − 1.84i·13-s + 16-s + 0.941i·17-s + i·18-s + (−0.600 − 0.800i)20-s + (−0.280 + 0.959i)25-s − 1.84·26-s + 1.44·29-s i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.599 + 0.799i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.599 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.599 + 0.799i$
Analytic conductor: \(0.544960\)
Root analytic conductor: \(0.738214\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :1),\ 0.599 + 0.799i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.741446 - 0.370723i\)
\(L(\frac12)\) \(\approx\) \(0.741446 - 0.370723i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
5 \( 1 + (-3 - 4i)T \)
good3 \( 1 + 9T^{2} \)
7 \( 1 + 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 24iT - 169T^{2} \)
17 \( 1 - 16iT - 289T^{2} \)
19 \( 1 - 361T^{2} \)
23 \( 1 + 529T^{2} \)
29 \( 1 - 42T + 841T^{2} \)
31 \( 1 - 961T^{2} \)
37 \( 1 + 24iT - 1.36e3T^{2} \)
41 \( 1 + 18T + 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 + 2.20e3T^{2} \)
53 \( 1 - 56iT - 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 - 22T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 96iT - 5.32e3T^{2} \)
79 \( 1 - 6.24e3T^{2} \)
83 \( 1 + 6.88e3T^{2} \)
89 \( 1 + 78T + 7.92e3T^{2} \)
97 \( 1 + 144iT - 9.40e3T^{2} \)
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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

−17.96062590567325815104023523697, −17.33399256142956227684504519776, −15.02155578445027214922818563134, −13.91615602008035128526283404749, −12.62713949989021404985482988469, −11.02542479449272167448992573251, −10.09372891876765178042217088505, −8.324070771202873362775595250013, −5.70955971770895197519786291105, −2.98280952584571453573837281815, 4.87586661189760822228494762830, 6.50419717571286079875050761495, 8.520031008385668286043288259253, 9.547024869041151195264697437329, 11.90550244298984676775472499594, 13.59782577234706309083810554246, 14.34275859722948862111176454258, 16.12067171012368066821763565845, 16.87855723835633671836689974348, 17.94885027651970621775195863569

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