Properties

Degree $2$
Conductor $5$
Sign $0.0306 - 0.999i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.20e4 + 4.20e4i)2-s + (−2.23e7 − 2.23e7i)3-s + 7.62e8i·4-s + (−7.62e10 − 1.32e11i)5-s + 1.87e12·6-s + (−2.39e13 + 2.39e13i)7-s + (−2.12e14 − 2.12e14i)8-s − 8.55e14i·9-s + (8.75e15 + 2.35e15i)10-s − 3.17e16·11-s + (1.70e16 − 1.70e16i)12-s + (−5.32e17 − 5.32e17i)13-s − 2.01e18i·14-s + (−1.25e18 + 4.65e18i)15-s + 1.45e19·16-s + (4.10e19 − 4.10e19i)17-s + ⋯
L(s)  = 1  + (−0.641 + 0.641i)2-s + (−0.518 − 0.518i)3-s + 0.177i·4-s + (−0.499 − 0.866i)5-s + 0.665·6-s + (−0.721 + 0.721i)7-s + (−0.755 − 0.755i)8-s − 0.461i·9-s + (0.875 + 0.235i)10-s − 0.690·11-s + (0.0920 − 0.0920i)12-s + (−0.800 − 0.800i)13-s − 0.925i·14-s + (−0.190 + 0.708i)15-s + 0.791·16-s + (0.842 − 0.842i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0306 - 0.999i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.0306 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.0306 - 0.999i$
Motivic weight: \(32\)
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ 0.0306 - 0.999i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.1823529760\)
\(L(\frac12)\) \(\approx\) \(0.1823529760\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (7.62e10 + 1.32e11i)T \)
good2 \( 1 + (4.20e4 - 4.20e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (2.23e7 + 2.23e7i)T + 1.85e15iT^{2} \)
7 \( 1 + (2.39e13 - 2.39e13i)T - 1.10e27iT^{2} \)
11 \( 1 + 3.17e16T + 2.11e33T^{2} \)
13 \( 1 + (5.32e17 + 5.32e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (-4.10e19 + 4.10e19i)T - 2.36e39iT^{2} \)
19 \( 1 + 2.22e20iT - 8.31e40T^{2} \)
23 \( 1 + (6.22e21 + 6.22e21i)T + 3.76e43iT^{2} \)
29 \( 1 - 3.25e23iT - 6.26e46T^{2} \)
31 \( 1 - 1.21e22T + 5.29e47T^{2} \)
37 \( 1 + (1.53e25 - 1.53e25i)T - 1.52e50iT^{2} \)
41 \( 1 + 2.70e25T + 4.06e51T^{2} \)
43 \( 1 + (-9.54e25 - 9.54e25i)T + 1.86e52iT^{2} \)
47 \( 1 + (-2.53e26 + 2.53e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (1.49e27 + 1.49e27i)T + 1.50e55iT^{2} \)
59 \( 1 + 2.48e28iT - 4.64e56T^{2} \)
61 \( 1 - 3.17e28T + 1.35e57T^{2} \)
67 \( 1 + (3.16e27 - 3.16e27i)T - 2.71e58iT^{2} \)
71 \( 1 - 4.69e29T + 1.73e59T^{2} \)
73 \( 1 + (3.04e29 + 3.04e29i)T + 4.22e59iT^{2} \)
79 \( 1 - 1.79e30iT - 5.29e60T^{2} \)
83 \( 1 + (4.68e29 + 4.68e29i)T + 2.57e61iT^{2} \)
89 \( 1 - 2.97e31iT - 2.40e62T^{2} \)
97 \( 1 + (1.89e31 - 1.89e31i)T - 3.77e63iT^{2} \)
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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

−16.41367927458907299366571381704, −15.44207762038731909064913205435, −12.67556484855137989660319929890, −12.15933757192349503633969811890, −9.552623557423060161907078090631, −8.216147647652124879331814929009, −6.89401720925537780754810672576, −5.34959075517241860559537467105, −3.08362096922098686678761327556, −0.59642719337839519039370033244, 0.13558411464732156635736030620, 2.13623157535274191655076133761, 3.82860377516859546656039771106, 5.79012678112373894454549817175, 7.65144237621860320529491927046, 9.967404189868347357731598524315, 10.50233943595461738569563738591, 11.85823028968646577902770525857, 14.15363768640244492134143940274, 15.76019016393890681252317818184

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