Properties

Degree $2$
Conductor $5$
Sign $-0.991 + 0.128i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−7.35e4 + 7.35e4i)2-s + (−4.29e6 − 4.29e6i)3-s − 6.51e9i·4-s + (−1.18e11 + 9.62e10i)5-s + 6.31e11·6-s + (1.36e13 − 1.36e13i)7-s + (1.63e14 + 1.63e14i)8-s − 1.81e15i·9-s + (1.62e15 − 1.57e16i)10-s − 1.07e16·11-s + (−2.79e16 + 2.79e16i)12-s + (5.33e17 + 5.33e17i)13-s + 2.00e18i·14-s + (9.21e17 + 9.48e16i)15-s + 3.95e18·16-s + (1.80e19 − 1.80e19i)17-s + ⋯
L(s)  = 1  + (−1.12 + 1.12i)2-s + (−0.0997 − 0.0997i)3-s − 1.51i·4-s + (−0.775 + 0.630i)5-s + 0.223·6-s + (0.410 − 0.410i)7-s + (0.580 + 0.580i)8-s − 0.980i·9-s + (0.162 − 1.57i)10-s − 0.234·11-s + (−0.151 + 0.151i)12-s + (0.801 + 0.801i)13-s + 0.921i·14-s + (0.140 + 0.0144i)15-s + 0.214·16-s + (0.370 − 0.370i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.991 + 0.128i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.991 + 0.128i$
Motivic weight: \(32\)
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ -0.991 + 0.128i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(0.4270473993\)
\(L(\frac12)\) \(\approx\) \(0.4270473993\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.18e11 - 9.62e10i)T \)
good2 \( 1 + (7.35e4 - 7.35e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (4.29e6 + 4.29e6i)T + 1.85e15iT^{2} \)
7 \( 1 + (-1.36e13 + 1.36e13i)T - 1.10e27iT^{2} \)
11 \( 1 + 1.07e16T + 2.11e33T^{2} \)
13 \( 1 + (-5.33e17 - 5.33e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (-1.80e19 + 1.80e19i)T - 2.36e39iT^{2} \)
19 \( 1 - 4.66e20iT - 8.31e40T^{2} \)
23 \( 1 + (-7.85e20 - 7.85e20i)T + 3.76e43iT^{2} \)
29 \( 1 + 2.31e23iT - 6.26e46T^{2} \)
31 \( 1 - 3.24e23T + 5.29e47T^{2} \)
37 \( 1 + (-4.94e24 + 4.94e24i)T - 1.52e50iT^{2} \)
41 \( 1 + 1.24e26T + 4.06e51T^{2} \)
43 \( 1 + (1.83e25 + 1.83e25i)T + 1.86e52iT^{2} \)
47 \( 1 + (-3.45e26 + 3.45e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (3.32e27 + 3.32e27i)T + 1.50e55iT^{2} \)
59 \( 1 - 3.29e28iT - 4.64e56T^{2} \)
61 \( 1 - 1.89e28T + 1.35e57T^{2} \)
67 \( 1 + (1.76e29 - 1.76e29i)T - 2.71e58iT^{2} \)
71 \( 1 + 1.94e29T + 1.73e59T^{2} \)
73 \( 1 + (-3.08e29 - 3.08e29i)T + 4.22e59iT^{2} \)
79 \( 1 - 3.31e30iT - 5.29e60T^{2} \)
83 \( 1 + (6.01e30 + 6.01e30i)T + 2.57e61iT^{2} \)
89 \( 1 - 2.42e31iT - 2.40e62T^{2} \)
97 \( 1 + (-2.36e31 + 2.36e31i)T - 3.77e63iT^{2} \)
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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

−16.74930485130022040376998593662, −15.51426368255916133219630808700, −14.35822148897825025412509558572, −11.76349414516633949944355053923, −10.08964868450127659113507706264, −8.451771728370416737060363576010, −7.29497163737103243187658190114, −6.14501339695589436046240725445, −3.75394884008639128035696217205, −1.11173939324006904646133475240, 0.24008603235580834813120836065, 1.50687540882365609001345078377, 3.06771567480535353152258221374, 4.99608211693408588750416435581, 7.918016684838769778199612547953, 8.804087732987192936499727516869, 10.58803679019920168669879405685, 11.55840951845712066598571142027, 12.99256594414079634559286338335, 15.56265850723579025246015336905

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