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Properties

Label 2-20-20.19-c10-0-14
Degree $2$
Conductor $20$
Sign $0.528 + 0.849i$
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  + (21.8 + 23.3i)2-s − 416.·3-s + (−71.0 + 1.02e3i)4-s + (−2.53e3 + 1.83e3i)5-s + (−9.09e3 − 9.75e3i)6-s + 9.55e3·7-s + (−2.54e4 + 2.06e4i)8-s + 1.14e5·9-s + (−9.81e4 − 1.92e4i)10-s − 3.04e5i·11-s + (2.96e4 − 4.25e5i)12-s + 2.93e4i·13-s + (2.08e5 + 2.23e5i)14-s + (1.05e6 − 7.63e5i)15-s + (−1.03e6 − 1.45e5i)16-s + 1.45e6i·17-s + ⋯
L(s)  = 1  + (0.682 + 0.731i)2-s − 1.71·3-s + (−0.0694 + 0.997i)4-s + (−0.810 + 0.585i)5-s + (−1.16 − 1.25i)6-s + 0.568·7-s + (−0.776 + 0.629i)8-s + 1.94·9-s + (−0.981 − 0.192i)10-s − 1.89i·11-s + (0.119 − 1.71i)12-s + 0.0791i·13-s + (0.387 + 0.415i)14-s + (1.38 − 1.00i)15-s + (−0.990 − 0.138i)16-s + 1.02i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $0.528 + 0.849i$
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: \((2,\ 20,\ (\ :5),\ 0.528 + 0.849i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.362417 - 0.201337i\)
\(L(\frac12)\) \(\approx\) \(0.362417 - 0.201337i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-21.8 - 23.3i)T \)
5 \( 1 + (2.53e3 - 1.83e3i)T \)
good3 \( 1 + 416.T + 5.90e4T^{2} \)
7 \( 1 - 9.55e3T + 2.82e8T^{2} \)
11 \( 1 + 3.04e5iT - 2.59e10T^{2} \)
13 \( 1 - 2.93e4iT - 1.37e11T^{2} \)
17 \( 1 - 1.45e6iT - 2.01e12T^{2} \)
19 \( 1 + 6.37e5iT - 6.13e12T^{2} \)
23 \( 1 - 3.42e5T + 4.14e13T^{2} \)
29 \( 1 + 2.17e7T + 4.20e14T^{2} \)
31 \( 1 + 2.54e7iT - 8.19e14T^{2} \)
37 \( 1 + 6.66e7iT - 4.80e15T^{2} \)
41 \( 1 - 9.83e7T + 1.34e16T^{2} \)
43 \( 1 + 9.92e7T + 2.16e16T^{2} \)
47 \( 1 + 2.47e8T + 5.25e16T^{2} \)
53 \( 1 + 4.80e8iT - 1.74e17T^{2} \)
59 \( 1 + 4.28e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.43e9T + 7.13e17T^{2} \)
67 \( 1 - 1.31e8T + 1.82e18T^{2} \)
71 \( 1 + 1.02e8iT - 3.25e18T^{2} \)
73 \( 1 + 8.16e8iT - 4.29e18T^{2} \)
79 \( 1 - 1.66e9iT - 9.46e18T^{2} \)
83 \( 1 + 4.49e9T + 1.55e19T^{2} \)
89 \( 1 + 1.63e9T + 3.11e19T^{2} \)
97 \( 1 - 8.94e9iT - 7.37e19T^{2} \)
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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

−15.96482636708935400425629331866, −14.63356447445421724609008694937, −12.96496205148555048913524596837, −11.48820883869584800747903908913, −11.07608655631214961375197122699, −8.082925884524272629568514740615, −6.54125494258251860516430953787, −5.48223966586068067609534166862, −3.86349978864433143271737998540, −0.19293748825419004468164715976, 1.36538433527505682475453542726, 4.45179996958137590214846136553, 5.18866778680284178239879328808, 7.08297489846237649394435789975, 9.841930295574166678904775235057, 11.24270597043032056034598080564, 12.02290193992702748890492045928, 12.82869585809346425385907012207, 14.95728864211855483702455170440, 16.08782446278165813631853979240

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