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Properties

Label 2-20-20.19-c10-0-25
Degree $2$
Conductor $20$
Sign $0.402 + 0.915i$
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  + (28.1 − 15.1i)2-s + 349.·3-s + (565. − 853. i)4-s + (1.69e3 − 2.62e3i)5-s + (9.85e3 − 5.29e3i)6-s − 1.54e4·7-s + (3.01e3 − 3.26e4i)8-s + 6.31e4·9-s + (7.84e3 − 9.96e4i)10-s + 1.36e5i·11-s + (1.97e5 − 2.98e5i)12-s + 4.67e5i·13-s + (−4.34e5 + 2.33e5i)14-s + (5.90e5 − 9.18e5i)15-s + (−4.09e5 − 9.65e5i)16-s − 1.19e6i·17-s + ⋯
L(s)  = 1  + (0.880 − 0.473i)2-s + 1.43·3-s + (0.552 − 0.833i)4-s + (0.540 − 0.841i)5-s + (1.26 − 0.680i)6-s − 0.916·7-s + (0.0918 − 0.995i)8-s + 1.06·9-s + (0.0784 − 0.996i)10-s + 0.849i·11-s + (0.794 − 1.19i)12-s + 1.25i·13-s + (−0.807 + 0.433i)14-s + (0.778 − 1.21i)15-s + (−0.390 − 0.920i)16-s − 0.839i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(12.7071\)
Root analytic conductor: \(3.56470\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :5),\ 0.402 + 0.915i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(3.81883 - 2.49205i\)
\(L(\frac12)\) \(\approx\) \(3.81883 - 2.49205i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-28.1 + 15.1i)T \)
5 \( 1 + (-1.69e3 + 2.62e3i)T \)
good3 \( 1 - 349.T + 5.90e4T^{2} \)
7 \( 1 + 1.54e4T + 2.82e8T^{2} \)
11 \( 1 - 1.36e5iT - 2.59e10T^{2} \)
13 \( 1 - 4.67e5iT - 1.37e11T^{2} \)
17 \( 1 + 1.19e6iT - 2.01e12T^{2} \)
19 \( 1 - 1.42e6iT - 6.13e12T^{2} \)
23 \( 1 - 1.04e7T + 4.14e13T^{2} \)
29 \( 1 - 8.98e6T + 4.20e14T^{2} \)
31 \( 1 - 2.79e7iT - 8.19e14T^{2} \)
37 \( 1 - 1.06e8iT - 4.80e15T^{2} \)
41 \( 1 + 1.66e8T + 1.34e16T^{2} \)
43 \( 1 - 7.53e7T + 2.16e16T^{2} \)
47 \( 1 - 2.17e8T + 5.25e16T^{2} \)
53 \( 1 + 5.32e8iT - 1.74e17T^{2} \)
59 \( 1 + 2.43e6iT - 5.11e17T^{2} \)
61 \( 1 + 1.38e9T + 7.13e17T^{2} \)
67 \( 1 + 1.26e9T + 1.82e18T^{2} \)
71 \( 1 + 2.18e9iT - 3.25e18T^{2} \)
73 \( 1 + 5.11e8iT - 4.29e18T^{2} \)
79 \( 1 - 4.53e9iT - 9.46e18T^{2} \)
83 \( 1 + 8.51e8T + 1.55e19T^{2} \)
89 \( 1 - 2.46e9T + 3.11e19T^{2} \)
97 \( 1 + 8.02e9iT - 7.37e19T^{2} \)
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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

−15.30255866631206794029483624093, −14.05342588168051893393146691190, −13.29641187176174142027637736630, −12.17350205356848647143858417997, −9.876123892089200173174059274110, −9.040095897277613187203683009871, −6.83225412522901886252664903737, −4.69970425352648116058230131548, −3.06855188313917346240979831825, −1.66090845621004240729789412337, 2.69436735726454747554300523155, 3.41936686467700632778186693801, 5.97182670314446752100810786370, 7.45398738472026664741910317739, 8.930701597223056395974124536100, 10.69370887415110578601790725339, 12.96500631432415400141514552363, 13.63150529766953165332517083358, 14.80930776699684638395159252319, 15.53815521463306775037696024561

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