Properties

Degree $2$
Conductor $5$
Sign $0.874 + 0.484i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.17e4 + 6.17e4i)2-s + (−5.69e7 + 5.69e7i)3-s + 3.32e9i·4-s + (7.47e10 + 1.33e11i)5-s − 7.03e12·6-s + (−1.03e13 − 1.03e13i)7-s + (5.97e13 − 5.97e13i)8-s − 4.63e15i·9-s + (−3.59e15 + 1.28e16i)10-s − 8.55e16·11-s + (−1.89e17 − 1.89e17i)12-s + (−2.19e16 + 2.19e16i)13-s − 1.27e18i·14-s + (−1.18e19 − 3.32e18i)15-s + 2.16e19·16-s + (9.87e18 + 9.87e18i)17-s + ⋯
L(s)  = 1  + (0.942 + 0.942i)2-s + (−1.32 + 1.32i)3-s + 0.774i·4-s + (0.489 + 0.871i)5-s − 2.49·6-s + (−0.311 − 0.311i)7-s + (0.212 − 0.212i)8-s − 2.50i·9-s + (−0.359 + 1.28i)10-s − 1.86·11-s + (−1.02 − 1.02i)12-s + (−0.0329 + 0.0329i)13-s − 0.587i·14-s + (−1.80 − 0.505i)15-s + 1.17·16-s + (0.202 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.874 + 0.484i$
Motivic weight: \(32\)
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ 0.874 + 0.484i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.1655894908\)
\(L(\frac12)\) \(\approx\) \(0.1655894908\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-7.47e10 - 1.33e11i)T \)
good2 \( 1 + (-6.17e4 - 6.17e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (5.69e7 - 5.69e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (1.03e13 + 1.03e13i)T + 1.10e27iT^{2} \)
11 \( 1 + 8.55e16T + 2.11e33T^{2} \)
13 \( 1 + (2.19e16 - 2.19e16i)T - 4.42e35iT^{2} \)
17 \( 1 + (-9.87e18 - 9.87e18i)T + 2.36e39iT^{2} \)
19 \( 1 + 1.02e20iT - 8.31e40T^{2} \)
23 \( 1 + (-3.76e21 + 3.76e21i)T - 3.76e43iT^{2} \)
29 \( 1 - 1.14e23iT - 6.26e46T^{2} \)
31 \( 1 + 8.94e23T + 5.29e47T^{2} \)
37 \( 1 + (2.29e24 + 2.29e24i)T + 1.52e50iT^{2} \)
41 \( 1 + 1.78e25T + 4.06e51T^{2} \)
43 \( 1 + (4.64e25 - 4.64e25i)T - 1.86e52iT^{2} \)
47 \( 1 + (2.49e26 + 2.49e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (7.86e25 - 7.86e25i)T - 1.50e55iT^{2} \)
59 \( 1 + 1.58e28iT - 4.64e56T^{2} \)
61 \( 1 - 2.93e28T + 1.35e57T^{2} \)
67 \( 1 + (1.50e29 + 1.50e29i)T + 2.71e58iT^{2} \)
71 \( 1 + 4.66e29T + 1.73e59T^{2} \)
73 \( 1 + (7.79e27 - 7.79e27i)T - 4.22e59iT^{2} \)
79 \( 1 - 4.15e30iT - 5.29e60T^{2} \)
83 \( 1 + (-3.04e30 + 3.04e30i)T - 2.57e61iT^{2} \)
89 \( 1 + 2.29e31iT - 2.40e62T^{2} \)
97 \( 1 + (5.10e31 + 5.10e31i)T + 3.77e63iT^{2} \)
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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

−15.73914864563954469268569610265, −14.77741224839587962081558225203, −13.00374598760661952449235319192, −10.84153368309417860698807981181, −10.06677952766573678637472546181, −6.95484052242311363982697677054, −5.72386648925575793346954583312, −4.90786454301496047242999303726, −3.37251535385870762479495701441, −0.04375113521529425239173009420, 1.36013543266015232153485425643, 2.48839032299062724998272749832, 5.04909051401748269946287936486, 5.69894897086810633895513509434, 7.74908342540491224880322708336, 10.55276387276551913013643422315, 11.91379591239709929706744435576, 12.90585128478314441158338926074, 13.38795049647031745261469383771, 16.37413234117868495216819314889

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