Properties

Degree $2$
Conductor $5$
Sign $0.788 + 0.614i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.23e4 + 3.23e4i)2-s + (1.63e7 − 1.63e7i)3-s − 2.19e9i·4-s + (5.30e10 + 1.43e11i)5-s + 1.06e12·6-s + (−2.79e13 − 2.79e13i)7-s + (2.10e14 − 2.10e14i)8-s + 1.31e15i·9-s + (−2.91e15 + 6.35e15i)10-s + 6.58e16·11-s + (−3.59e16 − 3.59e16i)12-s + (−2.99e16 + 2.99e16i)13-s − 1.81e18i·14-s + (3.21e18 + 1.47e18i)15-s + 4.19e18·16-s + (−1.92e19 − 1.92e19i)17-s + ⋯
L(s)  = 1  + (0.494 + 0.494i)2-s + (0.380 − 0.380i)3-s − 0.511i·4-s + (0.347 + 0.937i)5-s + 0.375·6-s + (−0.842 − 0.842i)7-s + (0.747 − 0.747i)8-s + 0.710i·9-s + (−0.291 + 0.635i)10-s + 1.43·11-s + (−0.194 − 0.194i)12-s + (−0.0449 + 0.0449i)13-s − 0.832i·14-s + (0.488 + 0.224i)15-s + 0.227·16-s + (−0.396 − 0.396i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.788 + 0.614i$
Motivic weight: \(32\)
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ 0.788 + 0.614i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(3.138716398\)
\(L(\frac12)\) \(\approx\) \(3.138716398\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-5.30e10 - 1.43e11i)T \)
good2 \( 1 + (-3.23e4 - 3.23e4i)T + 4.29e9iT^{2} \)
3 \( 1 + (-1.63e7 + 1.63e7i)T - 1.85e15iT^{2} \)
7 \( 1 + (2.79e13 + 2.79e13i)T + 1.10e27iT^{2} \)
11 \( 1 - 6.58e16T + 2.11e33T^{2} \)
13 \( 1 + (2.99e16 - 2.99e16i)T - 4.42e35iT^{2} \)
17 \( 1 + (1.92e19 + 1.92e19i)T + 2.36e39iT^{2} \)
19 \( 1 + 5.51e20iT - 8.31e40T^{2} \)
23 \( 1 + (-3.28e21 + 3.28e21i)T - 3.76e43iT^{2} \)
29 \( 1 + 3.10e23iT - 6.26e46T^{2} \)
31 \( 1 - 1.32e24T + 5.29e47T^{2} \)
37 \( 1 + (-2.81e24 - 2.81e24i)T + 1.52e50iT^{2} \)
41 \( 1 - 3.62e25T + 4.06e51T^{2} \)
43 \( 1 + (-1.14e26 + 1.14e26i)T - 1.86e52iT^{2} \)
47 \( 1 + (7.62e24 + 7.62e24i)T + 3.21e53iT^{2} \)
53 \( 1 + (4.02e27 - 4.02e27i)T - 1.50e55iT^{2} \)
59 \( 1 + 3.60e27iT - 4.64e56T^{2} \)
61 \( 1 + 1.10e28T + 1.35e57T^{2} \)
67 \( 1 + (-5.94e27 - 5.94e27i)T + 2.71e58iT^{2} \)
71 \( 1 + 2.05e29T + 1.73e59T^{2} \)
73 \( 1 + (-4.18e29 + 4.18e29i)T - 4.22e59iT^{2} \)
79 \( 1 - 2.78e30iT - 5.29e60T^{2} \)
83 \( 1 + (6.45e30 - 6.45e30i)T - 2.57e61iT^{2} \)
89 \( 1 - 7.48e30iT - 2.40e62T^{2} \)
97 \( 1 + (1.14e31 + 1.14e31i)T + 3.77e63iT^{2} \)
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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

−15.47785571613728348180744184898, −14.02221014340266959725427761023, −13.46242383502305504578780460178, −10.94421031636573959568431587154, −9.558174285832953314661401517982, −7.08708537974987779075526455724, −6.44308299648567419539975710820, −4.38869193221304888404811125395, −2.60760726329035090137855558976, −0.849919195430915609795025968753, 1.46565126206361028908391978514, 3.16183863669476089529451481600, 4.24720150857309501973885078614, 6.11896995607835607551617736807, 8.559371495240113189945451146250, 9.571654048257548248517443280276, 11.98448795105539126660270227805, 12.71305295873375844805570419886, 14.37055732524040586379213222536, 16.13144577162996801018147823475

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