Properties

Degree $2$
Conductor $5$
Sign $0.0307 + 0.999i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.95e4 − 3.95e4i)2-s + (4.69e7 − 4.69e7i)3-s − 1.16e9i·4-s + (−7.61e10 + 1.32e11i)5-s − 3.71e12·6-s + (2.85e13 + 2.85e13i)7-s + (−2.16e14 + 2.16e14i)8-s − 2.55e15i·9-s + (8.23e15 − 2.21e15i)10-s + 6.29e16·11-s + (−5.48e16 − 5.48e16i)12-s + (6.68e17 − 6.68e17i)13-s − 2.25e18i·14-s + (2.63e18 + 9.78e18i)15-s + 1.20e19·16-s + (9.86e17 + 9.86e17i)17-s + ⋯
L(s)  = 1  + (−0.603 − 0.603i)2-s + (1.09 − 1.09i)3-s − 0.272i·4-s + (−0.499 + 0.866i)5-s − 1.31·6-s + (0.858 + 0.858i)7-s + (−0.767 + 0.767i)8-s − 1.37i·9-s + (0.823 − 0.221i)10-s + 1.36·11-s + (−0.296 − 0.296i)12-s + (1.00 − 1.00i)13-s − 1.03i·14-s + (0.400 + 1.48i)15-s + 0.653·16-s + (0.0202 + 0.0202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0307 + 0.999i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.0307 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.0307 + 0.999i$
Motivic weight: \(32\)
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ 0.0307 + 0.999i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(2.374294459\)
\(L(\frac12)\) \(\approx\) \(2.374294459\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (7.61e10 - 1.32e11i)T \)
good2 \( 1 + (3.95e4 + 3.95e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (-4.69e7 + 4.69e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (-2.85e13 - 2.85e13i)T + 1.10e27iT^{2} \)
11 \( 1 - 6.29e16T + 2.11e33T^{2} \)
13 \( 1 + (-6.68e17 + 6.68e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (-9.86e17 - 9.86e17i)T + 2.36e39iT^{2} \)
19 \( 1 - 3.77e20iT - 8.31e40T^{2} \)
23 \( 1 + (-5.92e21 + 5.92e21i)T - 3.76e43iT^{2} \)
29 \( 1 + 2.77e19iT - 6.26e46T^{2} \)
31 \( 1 + 3.93e23T + 5.29e47T^{2} \)
37 \( 1 + (4.19e24 + 4.19e24i)T + 1.52e50iT^{2} \)
41 \( 1 - 8.53e25T + 4.06e51T^{2} \)
43 \( 1 + (8.89e25 - 8.89e25i)T - 1.86e52iT^{2} \)
47 \( 1 + (-3.07e26 - 3.07e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (-4.54e26 + 4.54e26i)T - 1.50e55iT^{2} \)
59 \( 1 + 1.68e28iT - 4.64e56T^{2} \)
61 \( 1 + 1.17e28T + 1.35e57T^{2} \)
67 \( 1 + (1.91e29 + 1.91e29i)T + 2.71e58iT^{2} \)
71 \( 1 + 1.41e29T + 1.73e59T^{2} \)
73 \( 1 + (-5.53e29 + 5.53e29i)T - 4.22e59iT^{2} \)
79 \( 1 - 1.90e30iT - 5.29e60T^{2} \)
83 \( 1 + (-2.78e30 + 2.78e30i)T - 2.57e61iT^{2} \)
89 \( 1 + 2.46e30iT - 2.40e62T^{2} \)
97 \( 1 + (-4.72e31 - 4.72e31i)T + 3.77e63iT^{2} \)
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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

−14.91956353007382474742821786077, −14.33462198891936668488200246170, −12.23637273658500048664917437814, −10.93235250356633781986512347391, −8.930549038402896780723723736713, −7.973045833102725641035011370665, −6.20385437567295947610059120678, −3.26766756781750666141263797390, −2.03156842782430964661062216484, −1.04715098117995375637562745237, 1.13493373264332515013116975939, 3.64666913858317806463352838856, 4.39368386534366931055303787208, 7.25592639326110641666780781164, 8.722024969221625184090077291070, 9.178905802867691918955164154774, 11.41583079108673792590652169717, 13.67974019418091258020397434910, 15.14618346542925912493419386213, 16.31876585012075478731780234893

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