Properties

Degree $2$
Conductor $5$
Sign $-0.736 - 0.675i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.43e4 − 2.43e4i)2-s + (−2.67e7 − 2.67e7i)3-s + 3.10e9i·4-s + (−1.49e11 − 2.86e10i)5-s − 1.30e12·6-s + (4.33e13 − 4.33e13i)7-s + (1.80e14 + 1.80e14i)8-s − 4.16e14i·9-s + (−4.34e15 + 2.95e15i)10-s + 2.50e16·11-s + (8.33e16 − 8.33e16i)12-s + (−7.52e17 − 7.52e17i)13-s − 2.11e18i·14-s + (3.24e18 + 4.78e18i)15-s − 4.57e18·16-s + (−3.85e19 + 3.85e19i)17-s + ⋯
L(s)  = 1  + (0.371 − 0.371i)2-s + (−0.622 − 0.622i)3-s + 0.723i·4-s + (−0.982 − 0.187i)5-s − 0.462·6-s + (1.30 − 1.30i)7-s + (0.640 + 0.640i)8-s − 0.225i·9-s + (−0.434 + 0.295i)10-s + 0.545·11-s + (0.450 − 0.450i)12-s + (−1.13 − 1.13i)13-s − 0.970i·14-s + (0.494 + 0.728i)15-s − 0.247·16-s + (−0.791 + 0.791i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.736 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.736 - 0.675i$
Motivic weight: \(32\)
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ -0.736 - 0.675i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.3466168509\)
\(L(\frac12)\) \(\approx\) \(0.3466168509\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.49e11 + 2.86e10i)T \)
good2 \( 1 + (-2.43e4 + 2.43e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (2.67e7 + 2.67e7i)T + 1.85e15iT^{2} \)
7 \( 1 + (-4.33e13 + 4.33e13i)T - 1.10e27iT^{2} \)
11 \( 1 - 2.50e16T + 2.11e33T^{2} \)
13 \( 1 + (7.52e17 + 7.52e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (3.85e19 - 3.85e19i)T - 2.36e39iT^{2} \)
19 \( 1 - 1.17e20iT - 8.31e40T^{2} \)
23 \( 1 + (-3.24e21 - 3.24e21i)T + 3.76e43iT^{2} \)
29 \( 1 + 2.76e23iT - 6.26e46T^{2} \)
31 \( 1 + 7.57e23T + 5.29e47T^{2} \)
37 \( 1 + (6.38e24 - 6.38e24i)T - 1.52e50iT^{2} \)
41 \( 1 + 4.00e25T + 4.06e51T^{2} \)
43 \( 1 + (2.44e24 + 2.44e24i)T + 1.86e52iT^{2} \)
47 \( 1 + (7.35e26 - 7.35e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (-4.27e26 - 4.27e26i)T + 1.50e55iT^{2} \)
59 \( 1 + 1.79e28iT - 4.64e56T^{2} \)
61 \( 1 + 1.42e28T + 1.35e57T^{2} \)
67 \( 1 + (1.58e29 - 1.58e29i)T - 2.71e58iT^{2} \)
71 \( 1 - 2.34e29T + 1.73e59T^{2} \)
73 \( 1 + (1.18e29 + 1.18e29i)T + 4.22e59iT^{2} \)
79 \( 1 + 1.26e30iT - 5.29e60T^{2} \)
83 \( 1 + (-7.76e29 - 7.76e29i)T + 2.57e61iT^{2} \)
89 \( 1 - 4.78e30iT - 2.40e62T^{2} \)
97 \( 1 + (-1.96e31 + 1.96e31i)T - 3.77e63iT^{2} \)
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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

−14.81443253722381247530254082343, −13.04112516187158533013953029515, −11.89682778774975062062549646080, −10.97430427445397157762892517351, −8.052188519308367113242518169168, −7.21198501473181331900253397818, −4.76177842794674225395782851978, −3.64437956107839745968970393735, −1.45808526663774363478177513212, −0.10284374102567273169266187783, 1.98380255349361106604406654963, 4.59039383081295223687500595758, 5.12581957798328152390694237982, 7.01696953459463913158235869329, 9.015451143207605071122246890710, 10.98928961332518211645917708876, 11.83598314607415230700100438294, 14.43503106524048453467939103598, 15.24640343460517912971638825882, 16.46758015844560262194322889801

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