Properties

Degree $2$
Conductor $5$
Sign $-0.724 + 0.689i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.09e3 + 7.09e3i)2-s + (−4.70e7 − 4.70e7i)3-s + 4.19e9i·4-s + (−3.87e10 + 1.47e11i)5-s + 6.67e11·6-s + (−3.04e13 + 3.04e13i)7-s + (−6.02e13 − 6.02e13i)8-s + 2.57e15i·9-s + (−7.72e14 − 1.32e15i)10-s + 3.11e16·11-s + (1.97e17 − 1.97e17i)12-s + (8.11e17 + 8.11e17i)13-s − 4.32e17i·14-s + (8.76e18 − 5.12e18i)15-s − 1.71e19·16-s + (−5.99e19 + 5.99e19i)17-s + ⋯
L(s)  = 1  + (−0.108 + 0.108i)2-s + (−1.09 − 1.09i)3-s + 0.976i·4-s + (−0.253 + 0.967i)5-s + 0.236·6-s + (−0.915 + 0.915i)7-s + (−0.214 − 0.214i)8-s + 1.38i·9-s + (−0.0772 − 0.132i)10-s + 0.678·11-s + (1.06 − 1.06i)12-s + (1.21 + 1.21i)13-s − 0.198i·14-s + (1.33 − 0.779i)15-s − 0.930·16-s + (−1.23 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.724 + 0.689i$
Motivic weight: \(32\)
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ -0.724 + 0.689i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.3905142101\)
\(L(\frac12)\) \(\approx\) \(0.3905142101\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.87e10 - 1.47e11i)T \)
good2 \( 1 + (7.09e3 - 7.09e3i)T - 4.29e9iT^{2} \)
3 \( 1 + (4.70e7 + 4.70e7i)T + 1.85e15iT^{2} \)
7 \( 1 + (3.04e13 - 3.04e13i)T - 1.10e27iT^{2} \)
11 \( 1 - 3.11e16T + 2.11e33T^{2} \)
13 \( 1 + (-8.11e17 - 8.11e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (5.99e19 - 5.99e19i)T - 2.36e39iT^{2} \)
19 \( 1 + 3.24e19iT - 8.31e40T^{2} \)
23 \( 1 + (-7.07e20 - 7.07e20i)T + 3.76e43iT^{2} \)
29 \( 1 - 1.29e23iT - 6.26e46T^{2} \)
31 \( 1 - 1.94e23T + 5.29e47T^{2} \)
37 \( 1 + (1.30e25 - 1.30e25i)T - 1.52e50iT^{2} \)
41 \( 1 - 1.97e25T + 4.06e51T^{2} \)
43 \( 1 + (1.49e26 + 1.49e26i)T + 1.86e52iT^{2} \)
47 \( 1 + (1.46e25 - 1.46e25i)T - 3.21e53iT^{2} \)
53 \( 1 + (-1.61e26 - 1.61e26i)T + 1.50e55iT^{2} \)
59 \( 1 + 1.65e28iT - 4.64e56T^{2} \)
61 \( 1 - 3.79e28T + 1.35e57T^{2} \)
67 \( 1 + (5.62e27 - 5.62e27i)T - 2.71e58iT^{2} \)
71 \( 1 - 2.56e29T + 1.73e59T^{2} \)
73 \( 1 + (-2.59e28 - 2.59e28i)T + 4.22e59iT^{2} \)
79 \( 1 + 6.46e29iT - 5.29e60T^{2} \)
83 \( 1 + (-5.85e30 - 5.85e30i)T + 2.57e61iT^{2} \)
89 \( 1 + 1.03e31iT - 2.40e62T^{2} \)
97 \( 1 + (-4.86e31 + 4.86e31i)T - 3.77e63iT^{2} \)
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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

−17.12131302765545473759569285227, −15.72345060658732072669706659613, −13.36894772825381475390113188397, −12.13481523668164772160826436657, −11.23324174653847646758129196113, −8.691425502513242175911901940777, −6.71707311773534381713326419044, −6.41454359253828709871233862550, −3.64772876669741944271599690016, −1.94641505677362411333241846732, 0.19851629527802802840232879137, 0.857489745379136954736464064284, 3.91011796903652939556086215076, 5.10769369222516075374482509527, 6.35808296419895822607684435518, 9.183742925670322821644030654431, 10.31597838112529023975336026577, 11.41066244437791222398129145112, 13.37792195221756647752007696070, 15.60045934119855186567837526130

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