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Properties

Label 2-5-5.2-c20-0-4
Degree $2$
Conductor $5$
Sign $0.387 - 0.921i$
Analytic cond. $12.6756$
Root an. cond. $3.56029$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (782. + 782. i)2-s + (−1.28e4 + 1.28e4i)3-s + 1.77e5i·4-s + (7.95e6 − 5.66e6i)5-s − 2.00e7·6-s + (3.15e8 + 3.15e8i)7-s + (6.82e8 − 6.82e8i)8-s + 3.15e9i·9-s + (1.06e10 + 1.79e9i)10-s − 2.67e10·11-s + (−2.26e9 − 2.26e9i)12-s + (9.28e9 − 9.28e9i)13-s + 4.93e11i·14-s + (−2.93e10 + 1.74e11i)15-s + 1.25e12·16-s + (2.62e12 + 2.62e12i)17-s + ⋯
L(s)  = 1  + (0.764 + 0.764i)2-s + (−0.216 + 0.216i)3-s + 0.168i·4-s + (0.814 − 0.580i)5-s − 0.331·6-s + (1.11 + 1.11i)7-s + (0.635 − 0.635i)8-s + 0.906i·9-s + (1.06 + 0.179i)10-s − 1.03·11-s + (−0.0366 − 0.0366i)12-s + (0.0673 − 0.0673i)13-s + 1.70i·14-s + (−0.0508 + 0.302i)15-s + 1.14·16-s + (1.30 + 1.30i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.387 - 0.921i$
Analytic conductor: \(12.6756\)
Root analytic conductor: \(3.56029\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :10),\ 0.387 - 0.921i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(2.60878 + 1.73229i\)
\(L(\frac12)\) \(\approx\) \(2.60878 + 1.73229i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-7.95e6 + 5.66e6i)T \)
good2 \( 1 + (-782. - 782. i)T + 1.04e6iT^{2} \)
3 \( 1 + (1.28e4 - 1.28e4i)T - 3.48e9iT^{2} \)
7 \( 1 + (-3.15e8 - 3.15e8i)T + 7.97e16iT^{2} \)
11 \( 1 + 2.67e10T + 6.72e20T^{2} \)
13 \( 1 + (-9.28e9 + 9.28e9i)T - 1.90e22iT^{2} \)
17 \( 1 + (-2.62e12 - 2.62e12i)T + 4.06e24iT^{2} \)
19 \( 1 + 1.22e12iT - 3.75e25T^{2} \)
23 \( 1 + (-1.74e13 + 1.74e13i)T - 1.71e27iT^{2} \)
29 \( 1 + 2.30e13iT - 1.76e29T^{2} \)
31 \( 1 + 9.47e13T + 6.71e29T^{2} \)
37 \( 1 + (3.89e15 + 3.89e15i)T + 2.31e31iT^{2} \)
41 \( 1 + 2.23e16T + 1.80e32T^{2} \)
43 \( 1 + (-1.30e16 + 1.30e16i)T - 4.67e32iT^{2} \)
47 \( 1 + (1.21e16 + 1.21e16i)T + 2.76e33iT^{2} \)
53 \( 1 + (9.59e16 - 9.59e16i)T - 3.05e34iT^{2} \)
59 \( 1 + 9.18e17iT - 2.61e35T^{2} \)
61 \( 1 + 6.23e16T + 5.08e35T^{2} \)
67 \( 1 + (4.75e17 + 4.75e17i)T + 3.32e36iT^{2} \)
71 \( 1 - 2.91e18T + 1.05e37T^{2} \)
73 \( 1 + (3.68e18 - 3.68e18i)T - 1.84e37iT^{2} \)
79 \( 1 - 2.24e18iT - 8.96e37T^{2} \)
83 \( 1 + (7.91e18 - 7.91e18i)T - 2.40e38iT^{2} \)
89 \( 1 + 1.81e18iT - 9.72e38T^{2} \)
97 \( 1 + (-2.60e19 - 2.60e19i)T + 5.43e39iT^{2} \)
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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

−18.74362162285391046810150328088, −17.01553519489451870133355480109, −15.57020279465654029394110079968, −14.22810855171721596211287393439, −12.74826156560639813999917852378, −10.44317817484210961932388973771, −8.174468327556595861677551135719, −5.62475522940624530613305983578, −5.01567745325729183638105029796, −1.81168490950033558138649206795, 1.35579833420532458318197378637, 3.18076165367505940583305706662, 5.12638268766302884432384910395, 7.45643317735524370398309379343, 10.33089645499576554045002567044, 11.67161426258776424702774509369, 13.41170505774397745411536425876, 14.44090841452566555812386019717, 17.11636694956027563568950732537, 18.29038363689814070400441077181

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