Properties

Degree $2$
Conductor $5$
Sign $0.992 + 0.124i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.66e4 − 8.66e4i)2-s + (3.32e7 − 3.32e7i)3-s + 1.07e10i·4-s + (1.18e11 + 9.57e10i)5-s − 5.76e12·6-s + (−3.06e13 − 3.06e13i)7-s + (5.57e14 − 5.57e14i)8-s − 3.55e14i·9-s + (−2.00e15 − 1.85e16i)10-s − 5.53e16·11-s + (3.56e17 + 3.56e17i)12-s + (1.61e17 − 1.61e17i)13-s + 5.31e18i·14-s + (7.12e18 − 7.66e17i)15-s − 5.06e19·16-s + (−2.56e19 − 2.56e19i)17-s + ⋯
L(s)  = 1  + (−1.32 − 1.32i)2-s + (0.771 − 0.771i)3-s + 2.49i·4-s + (0.778 + 0.627i)5-s − 2.04·6-s + (−0.922 − 0.922i)7-s + (1.98 − 1.98i)8-s − 0.191i·9-s + (−0.200 − 1.85i)10-s − 1.20·11-s + (1.92 + 1.92i)12-s + (0.243 − 0.243i)13-s + 2.43i·14-s + (1.08 − 0.116i)15-s − 2.74·16-s + (−0.527 − 0.527i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.992 + 0.124i$
Motivic weight: \(32\)
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ 0.992 + 0.124i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.7719159706\)
\(L(\frac12)\) \(\approx\) \(0.7719159706\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.18e11 - 9.57e10i)T \)
good2 \( 1 + (8.66e4 + 8.66e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (-3.32e7 + 3.32e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (3.06e13 + 3.06e13i)T + 1.10e27iT^{2} \)
11 \( 1 + 5.53e16T + 2.11e33T^{2} \)
13 \( 1 + (-1.61e17 + 1.61e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (2.56e19 + 2.56e19i)T + 2.36e39iT^{2} \)
19 \( 1 - 4.51e20iT - 8.31e40T^{2} \)
23 \( 1 + (-2.16e21 + 2.16e21i)T - 3.76e43iT^{2} \)
29 \( 1 - 1.69e23iT - 6.26e46T^{2} \)
31 \( 1 - 6.36e23T + 5.29e47T^{2} \)
37 \( 1 + (-1.46e24 - 1.46e24i)T + 1.52e50iT^{2} \)
41 \( 1 + 4.24e25T + 4.06e51T^{2} \)
43 \( 1 + (1.43e26 - 1.43e26i)T - 1.86e52iT^{2} \)
47 \( 1 + (-4.39e26 - 4.39e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (-1.94e27 + 1.94e27i)T - 1.50e55iT^{2} \)
59 \( 1 - 2.69e28iT - 4.64e56T^{2} \)
61 \( 1 - 1.09e28T + 1.35e57T^{2} \)
67 \( 1 + (-2.17e28 - 2.17e28i)T + 2.71e58iT^{2} \)
71 \( 1 - 5.41e29T + 1.73e59T^{2} \)
73 \( 1 + (-8.08e28 + 8.08e28i)T - 4.22e59iT^{2} \)
79 \( 1 - 1.28e30iT - 5.29e60T^{2} \)
83 \( 1 + (3.95e30 - 3.95e30i)T - 2.57e61iT^{2} \)
89 \( 1 - 6.87e30iT - 2.40e62T^{2} \)
97 \( 1 + (3.25e31 + 3.25e31i)T + 3.77e63iT^{2} \)
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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

−16.58376316301344577216868670428, −13.64293210339163537188431102734, −12.81375837946358549323098676127, −10.67699527948051945052544307754, −9.867597942890053345685478706099, −8.218408962512292950628445355065, −7.03161514748704233929472644640, −3.25040512177457504252007149520, −2.41924351855459688831233944784, −1.16050861565893199520108462473, 0.35797871908249982144063478598, 2.39554685029302205977730674815, 5.14940477876617655534409213101, 6.46778330276938746802644880357, 8.512290310887210164883505416501, 9.206317635078702780272142196517, 10.16492115081353920885099268657, 13.46645820485725397548604766369, 15.35723277808461418867379556920, 15.76766657003408675129090551596

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