Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.45e4 + 8.45e4i)2-s + (5.85e6 − 5.85e6i)3-s + 9.99e9i·4-s + (−1.33e11 + 7.46e10i)5-s + 9.89e11·6-s + (−4.06e12 − 4.06e12i)7-s + (−4.82e14 + 4.82e14i)8-s + 1.78e15i·9-s + (−1.75e16 − 4.93e15i)10-s − 1.93e16·11-s + (5.85e16 + 5.85e16i)12-s + (6.35e17 − 6.35e17i)13-s − 6.86e17i·14-s + (−3.41e17 + 1.21e18i)15-s − 3.86e19·16-s + (−4.90e19 − 4.90e19i)17-s + ⋯
L(s)  = 1  + (1.29 + 1.29i)2-s + (0.135 − 0.135i)3-s + 2.32i·4-s + (−0.872 + 0.489i)5-s + 0.350·6-s + (−0.122 − 0.122i)7-s + (−1.71 + 1.71i)8-s + 0.963i·9-s + (−1.75 − 0.493i)10-s − 0.421·11-s + (0.316 + 0.316i)12-s + (0.954 − 0.954i)13-s − 0.315i·14-s + (−0.0520 + 0.185i)15-s − 2.09·16-s + (−1.00 − 1.00i)17-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(1.883990593\)
\(L(\frac12)\) \(\approx\) \(1.883990593\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.33e11 - 7.46e10i)T \)
good2 \( 1 + (-8.45e4 - 8.45e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (-5.85e6 + 5.85e6i)T - 1.85e15iT^{2} \)
7 \( 1 + (4.06e12 + 4.06e12i)T + 1.10e27iT^{2} \)
11 \( 1 + 1.93e16T + 2.11e33T^{2} \)
13 \( 1 + (-6.35e17 + 6.35e17i)T - 4.42e35iT^{2} \)
17 \( 1 + (4.90e19 + 4.90e19i)T + 2.36e39iT^{2} \)
19 \( 1 - 5.30e17iT - 8.31e40T^{2} \)
23 \( 1 + (7.61e21 - 7.61e21i)T - 3.76e43iT^{2} \)
29 \( 1 + 1.82e21iT - 6.26e46T^{2} \)
31 \( 1 + 1.01e24T + 5.29e47T^{2} \)
37 \( 1 + (-3.94e24 - 3.94e24i)T + 1.52e50iT^{2} \)
41 \( 1 - 7.58e25T + 4.06e51T^{2} \)
43 \( 1 + (1.10e26 - 1.10e26i)T - 1.86e52iT^{2} \)
47 \( 1 + (-6.32e26 - 6.32e26i)T + 3.21e53iT^{2} \)
53 \( 1 + (3.02e27 - 3.02e27i)T - 1.50e55iT^{2} \)
59 \( 1 - 1.06e28iT - 4.64e56T^{2} \)
61 \( 1 + 3.36e27T + 1.35e57T^{2} \)
67 \( 1 + (-5.01e28 - 5.01e28i)T + 2.71e58iT^{2} \)
71 \( 1 - 6.84e28T + 1.73e59T^{2} \)
73 \( 1 + (8.59e29 - 8.59e29i)T - 4.22e59iT^{2} \)
79 \( 1 - 3.56e30iT - 5.29e60T^{2} \)
83 \( 1 + (1.63e30 - 1.63e30i)T - 2.57e61iT^{2} \)
89 \( 1 - 7.81e30iT - 2.40e62T^{2} \)
97 \( 1 + (3.43e31 + 3.43e31i)T + 3.77e63iT^{2} \)
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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

−16.15526269887831861059438097302, −15.51965725186397920074340177694, −13.99670245909888383168010671415, −12.97264648978423556354132845505, −11.20362267737574428716131629085, −8.107357391849486713730757368944, −7.24764831656578869566534331818, −5.65514658432825439339975024310, −4.19890002210710292321622820898, −2.90065116674033460761738151293, 0.34867497899820192131221733005, 1.90544090578039956125143896805, 3.63016931846890691512818374135, 4.34369203476592631585525830025, 6.17013142959846120277425629489, 8.926067219492064362343135459591, 10.79082069112052078179818942469, 12.00121494772120969440008696666, 13.00903930830784835234146771663, 14.59240117154027347007583315965

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