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Properties

Label 2-20-20.7-c7-0-9
Degree $2$
Conductor $20$
Sign $-0.0867 + 0.996i$
Analytic cond. $6.24770$
Root an. cond. $2.49954$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.3 − 0.411i)2-s + (−2.83 − 2.83i)3-s + (127. + 9.30i)4-s + (−150. + 235. i)5-s + (30.8 + 33.2i)6-s + (411. − 411. i)7-s + (−1.43e3 − 157. i)8-s − 2.17e3i·9-s + (1.79e3 − 2.60e3i)10-s − 7.80e3i·11-s + (−335. − 388. i)12-s + (−2.23e3 + 2.23e3i)13-s + (−4.82e3 + 4.48e3i)14-s + (1.09e3 − 241. i)15-s + (1.62e4 + 2.37e3i)16-s + (−1.85e4 − 1.85e4i)17-s + ⋯
L(s)  = 1  + (−0.999 − 0.0363i)2-s + (−0.0605 − 0.0605i)3-s + (0.997 + 0.0726i)4-s + (−0.537 + 0.843i)5-s + (0.0583 + 0.0627i)6-s + (0.453 − 0.453i)7-s + (−0.994 − 0.108i)8-s − 0.992i·9-s + (0.567 − 0.823i)10-s − 1.76i·11-s + (−0.0560 − 0.0648i)12-s + (−0.282 + 0.282i)13-s + (−0.469 + 0.436i)14-s + (0.0836 − 0.0185i)15-s + (0.989 + 0.144i)16-s + (−0.917 − 0.917i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0867 + 0.996i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0867 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(20\)    =    \(2^{2} \cdot 5\)
Sign: $-0.0867 + 0.996i$
Analytic conductor: \(6.24770\)
Root analytic conductor: \(2.49954\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{20} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 20,\ (\ :7/2),\ -0.0867 + 0.996i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.471770 - 0.514614i\)
\(L(\frac12)\) \(\approx\) \(0.471770 - 0.514614i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (11.3 + 0.411i)T \)
5 \( 1 + (150. - 235. i)T \)
good3 \( 1 + (2.83 + 2.83i)T + 2.18e3iT^{2} \)
7 \( 1 + (-411. + 411. i)T - 8.23e5iT^{2} \)
11 \( 1 + 7.80e3iT - 1.94e7T^{2} \)
13 \( 1 + (2.23e3 - 2.23e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (1.85e4 + 1.85e4i)T + 4.10e8iT^{2} \)
19 \( 1 - 3.48e4T + 8.93e8T^{2} \)
23 \( 1 + (-7.26e3 - 7.26e3i)T + 3.40e9iT^{2} \)
29 \( 1 - 3.55e4iT - 1.72e10T^{2} \)
31 \( 1 + 2.01e5iT - 2.75e10T^{2} \)
37 \( 1 + (-8.09e4 - 8.09e4i)T + 9.49e10iT^{2} \)
41 \( 1 - 5.23e4T + 1.94e11T^{2} \)
43 \( 1 + (2.50e5 + 2.50e5i)T + 2.71e11iT^{2} \)
47 \( 1 + (8.90e5 - 8.90e5i)T - 5.06e11iT^{2} \)
53 \( 1 + (-3.54e5 + 3.54e5i)T - 1.17e12iT^{2} \)
59 \( 1 + 8.58e5T + 2.48e12T^{2} \)
61 \( 1 + 1.30e6T + 3.14e12T^{2} \)
67 \( 1 + (-1.97e6 + 1.97e6i)T - 6.06e12iT^{2} \)
71 \( 1 + 4.95e6iT - 9.09e12T^{2} \)
73 \( 1 + (-6.38e5 + 6.38e5i)T - 1.10e13iT^{2} \)
79 \( 1 - 5.03e5T + 1.92e13T^{2} \)
83 \( 1 + (-4.22e6 - 4.22e6i)T + 2.71e13iT^{2} \)
89 \( 1 - 8.63e6iT - 4.42e13T^{2} \)
97 \( 1 + (3.20e6 + 3.20e6i)T + 8.07e13iT^{2} \)
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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

−16.49827627524703335051736785910, −15.36359711801672499924145114791, −13.99921958443826363837567025834, −11.67703097703449945028210274720, −11.03929599334042227400853615140, −9.327425559524357334858110578144, −7.77038873207292692499774924549, −6.43411442921747218471872623859, −3.21153794865915822403612172332, −0.55141608227124392636806584614, 1.83635972268798281184707076340, 4.97692990398896048484102946933, 7.38916763925937872559329852279, 8.552986669502401809295541900740, 10.02992991984800954012145152610, 11.55874493411957236653332383705, 12.72535204177583880547126119823, 14.99990780099603555838174528730, 15.94000181836583339966120525326, 17.17872985558875100902900378311

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