Properties

Degree $2$
Conductor $5$
Sign $0.888 + 0.459i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.10e3 + 5.10e3i)2-s + (−7.18e6 − 7.18e6i)3-s + 4.24e9i·4-s + (1.52e11 − 1.16e10i)5-s + 7.32e10·6-s + (2.17e13 − 2.17e13i)7-s + (−4.35e13 − 4.35e13i)8-s − 1.74e15i·9-s + (−7.16e14 + 8.35e14i)10-s − 5.19e16·11-s + (3.04e16 − 3.04e16i)12-s + (6.04e17 + 6.04e17i)13-s + 2.21e17i·14-s + (−1.17e18 − 1.00e18i)15-s − 1.77e19·16-s + (2.55e19 − 2.55e19i)17-s + ⋯
L(s)  = 1  + (−0.0778 + 0.0778i)2-s + (−0.166 − 0.166i)3-s + 0.987i·4-s + (0.997 − 0.0764i)5-s + 0.0259·6-s + (0.653 − 0.653i)7-s + (−0.154 − 0.154i)8-s − 0.944i·9-s + (−0.0716 + 0.0835i)10-s − 1.13·11-s + (0.164 − 0.164i)12-s + (0.907 + 0.907i)13-s + 0.101i·14-s + (−0.179 − 0.153i)15-s − 0.963·16-s + (0.524 − 0.524i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.888 + 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.888 + 0.459i$
Motivic weight: \(32\)
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ 0.888 + 0.459i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(2.211175145\)
\(L(\frac12)\) \(\approx\) \(2.211175145\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.52e11 + 1.16e10i)T \)
good2 \( 1 + (5.10e3 - 5.10e3i)T - 4.29e9iT^{2} \)
3 \( 1 + (7.18e6 + 7.18e6i)T + 1.85e15iT^{2} \)
7 \( 1 + (-2.17e13 + 2.17e13i)T - 1.10e27iT^{2} \)
11 \( 1 + 5.19e16T + 2.11e33T^{2} \)
13 \( 1 + (-6.04e17 - 6.04e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (-2.55e19 + 2.55e19i)T - 2.36e39iT^{2} \)
19 \( 1 + 1.62e20iT - 8.31e40T^{2} \)
23 \( 1 + (2.79e21 + 2.79e21i)T + 3.76e43iT^{2} \)
29 \( 1 + 3.52e23iT - 6.26e46T^{2} \)
31 \( 1 - 4.98e23T + 5.29e47T^{2} \)
37 \( 1 + (-1.32e25 + 1.32e25i)T - 1.52e50iT^{2} \)
41 \( 1 - 1.17e26T + 4.06e51T^{2} \)
43 \( 1 + (-9.57e25 - 9.57e25i)T + 1.86e52iT^{2} \)
47 \( 1 + (4.63e26 - 4.63e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (7.08e26 + 7.08e26i)T + 1.50e55iT^{2} \)
59 \( 1 - 1.33e28iT - 4.64e56T^{2} \)
61 \( 1 - 2.13e28T + 1.35e57T^{2} \)
67 \( 1 + (-2.29e29 + 2.29e29i)T - 2.71e58iT^{2} \)
71 \( 1 + 3.37e28T + 1.73e59T^{2} \)
73 \( 1 + (4.11e29 + 4.11e29i)T + 4.22e59iT^{2} \)
79 \( 1 - 9.94e29iT - 5.29e60T^{2} \)
83 \( 1 + (-1.82e30 - 1.82e30i)T + 2.57e61iT^{2} \)
89 \( 1 + 3.30e30iT - 2.40e62T^{2} \)
97 \( 1 + (-5.20e31 + 5.20e31i)T - 3.77e63iT^{2} \)
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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

−16.14482400052267070255025609452, −14.02577415603583880284745817597, −12.84673229761294820169770295740, −11.23558031307004616682511784808, −9.373170274500095469945295494129, −7.78496481260612859198695196432, −6.24617210671112776543657319156, −4.31159590815782959088594402218, −2.53990770475338674832202283236, −0.794943427505911133853280563008, 1.27122175707047454206767085485, 2.43348620295077394769335162215, 5.19526684384195570584260212193, 5.81406682060566338002063451438, 8.248161476369754246647610862910, 10.06956426739946515373938191042, 10.91780041174010233580945045689, 13.19423748486611198607685484744, 14.52571914865889949252366044175, 15.93429805934349669497120212348

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