Properties

Degree $2$
Conductor $5$
Sign $0.889 + 0.457i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.89e4 − 6.89e4i)2-s + (5.43e7 + 5.43e7i)3-s − 5.20e9i·4-s + (1.52e11 − 1.20e10i)5-s + 7.48e12·6-s + (3.40e13 − 3.40e13i)7-s + (−6.30e13 − 6.30e13i)8-s + 4.04e15i·9-s + (9.65e15 − 1.13e16i)10-s − 2.53e16·11-s + (2.82e17 − 2.82e17i)12-s + (−3.66e17 − 3.66e17i)13-s − 4.69e18i·14-s + (8.91e18 + 7.60e18i)15-s + 1.36e19·16-s + (−5.02e19 + 5.02e19i)17-s + ⋯
L(s)  = 1  + (1.05 − 1.05i)2-s + (1.26 + 1.26i)3-s − 1.21i·4-s + (0.996 − 0.0787i)5-s + 2.65·6-s + (1.02 − 1.02i)7-s + (−0.224 − 0.224i)8-s + 2.18i·9-s + (0.965 − 1.13i)10-s − 0.551·11-s + (1.53 − 1.53i)12-s + (−0.551 − 0.551i)13-s − 2.15i·14-s + (1.35 + 1.15i)15-s + 0.741·16-s + (−1.03 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.889 + 0.457i$
Motivic weight: \(32\)
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :16),\ 0.889 + 0.457i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(6.783146398\)
\(L(\frac12)\) \(\approx\) \(6.783146398\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.52e11 + 1.20e10i)T \)
good2 \( 1 + (-6.89e4 + 6.89e4i)T - 4.29e9iT^{2} \)
3 \( 1 + (-5.43e7 - 5.43e7i)T + 1.85e15iT^{2} \)
7 \( 1 + (-3.40e13 + 3.40e13i)T - 1.10e27iT^{2} \)
11 \( 1 + 2.53e16T + 2.11e33T^{2} \)
13 \( 1 + (3.66e17 + 3.66e17i)T + 4.42e35iT^{2} \)
17 \( 1 + (5.02e19 - 5.02e19i)T - 2.36e39iT^{2} \)
19 \( 1 + 6.86e19iT - 8.31e40T^{2} \)
23 \( 1 + (1.56e21 + 1.56e21i)T + 3.76e43iT^{2} \)
29 \( 1 + 6.33e22iT - 6.26e46T^{2} \)
31 \( 1 - 5.37e23T + 5.29e47T^{2} \)
37 \( 1 + (4.80e24 - 4.80e24i)T - 1.52e50iT^{2} \)
41 \( 1 + 7.84e25T + 4.06e51T^{2} \)
43 \( 1 + (-6.30e25 - 6.30e25i)T + 1.86e52iT^{2} \)
47 \( 1 + (-2.58e26 + 2.58e26i)T - 3.21e53iT^{2} \)
53 \( 1 + (4.72e27 + 4.72e27i)T + 1.50e55iT^{2} \)
59 \( 1 - 2.35e28iT - 4.64e56T^{2} \)
61 \( 1 + 2.89e28T + 1.35e57T^{2} \)
67 \( 1 + (6.00e28 - 6.00e28i)T - 2.71e58iT^{2} \)
71 \( 1 + 3.68e28T + 1.73e59T^{2} \)
73 \( 1 + (-1.90e29 - 1.90e29i)T + 4.22e59iT^{2} \)
79 \( 1 - 3.04e30iT - 5.29e60T^{2} \)
83 \( 1 + (-1.94e30 - 1.94e30i)T + 2.57e61iT^{2} \)
89 \( 1 + 2.88e31iT - 2.40e62T^{2} \)
97 \( 1 + (1.59e31 - 1.59e31i)T - 3.77e63iT^{2} \)
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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

−15.07409785052187230400209600743, −14.06373226880783867186080028916, −13.24762349289070477553042193604, −10.77283818861638466600968669958, −10.09543540935914333069176152768, −8.246503438705745099738645633983, −5.01952984003516179309867492833, −4.19943226620847429675390078995, −2.79134754662911298419299185633, −1.75946380887577114920208414982, 1.72873055876823573074643766867, 2.69347141804236883324572964102, 5.00482090463608627664072513311, 6.44413544748130003591270970868, 7.66888482799622440613002222427, 9.013674431654323688352927917511, 12.29921052768529585686189974372, 13.60694583676260591143980746200, 14.23923892939614306604343857163, 15.34285233205308040611395535373

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