Properties

Degree $2$
Conductor $4$
Sign $0.716 + 0.697i$
Motivic weight $32$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (6.07e4 + 2.46e4i)2-s − 5.10e7i·3-s + (3.07e9 + 2.99e9i)4-s + 2.49e11·5-s + (1.25e12 − 3.09e12i)6-s − 4.65e13i·7-s + (1.12e14 + 2.57e14i)8-s − 7.49e14·9-s + (1.51e16 + 6.15e15i)10-s + 4.19e16i·11-s + (1.52e17 − 1.56e17i)12-s − 6.87e17·13-s + (1.14e18 − 2.82e18i)14-s − 1.27e19i·15-s + (4.88e17 + 1.84e19i)16-s + 4.91e19·17-s + ⋯
L(s)  = 1  + (0.926 + 0.376i)2-s − 1.18i·3-s + (0.716 + 0.697i)4-s + 1.63·5-s + (0.446 − 1.09i)6-s − 1.39i·7-s + (0.400 + 0.916i)8-s − 0.404·9-s + (1.51 + 0.615i)10-s + 0.912i·11-s + (0.826 − 0.849i)12-s − 1.03·13-s + (0.526 − 1.29i)14-s − 1.93i·15-s + (0.0264 + 0.999i)16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.716 + 0.697i$
Motivic weight: \(32\)
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :16),\ 0.716 + 0.697i)\)

Particular Values

\(L(\frac{33}{2})\) \(\approx\) \(4.659242403\)
\(L(\frac12)\) \(\approx\) \(4.659242403\)
\(L(17)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-6.07e4 - 2.46e4i)T \)
good3 \( 1 + 5.10e7iT - 1.85e15T^{2} \)
5 \( 1 - 2.49e11T + 2.32e22T^{2} \)
7 \( 1 + 4.65e13iT - 1.10e27T^{2} \)
11 \( 1 - 4.19e16iT - 2.11e33T^{2} \)
13 \( 1 + 6.87e17T + 4.42e35T^{2} \)
17 \( 1 - 4.91e19T + 2.36e39T^{2} \)
19 \( 1 + 3.14e20iT - 8.31e40T^{2} \)
23 \( 1 + 4.04e21iT - 3.76e43T^{2} \)
29 \( 1 - 1.16e23T + 6.26e46T^{2} \)
31 \( 1 - 3.53e23iT - 5.29e47T^{2} \)
37 \( 1 + 7.58e24T + 1.52e50T^{2} \)
41 \( 1 + 2.78e25T + 4.06e51T^{2} \)
43 \( 1 - 1.82e26iT - 1.86e52T^{2} \)
47 \( 1 + 6.77e25iT - 3.21e53T^{2} \)
53 \( 1 - 1.13e27T + 1.50e55T^{2} \)
59 \( 1 - 5.33e27iT - 4.64e56T^{2} \)
61 \( 1 + 5.47e28T + 1.35e57T^{2} \)
67 \( 1 - 6.20e28iT - 2.71e58T^{2} \)
71 \( 1 - 4.47e29iT - 1.73e59T^{2} \)
73 \( 1 - 5.03e29T + 4.22e59T^{2} \)
79 \( 1 - 1.79e30iT - 5.29e60T^{2} \)
83 \( 1 + 2.17e30iT - 2.57e61T^{2} \)
89 \( 1 - 1.19e31T + 2.40e62T^{2} \)
97 \( 1 + 9.02e31T + 3.77e63T^{2} \)
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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

−16.97169834091400908535901477491, −14.36517615578230529195099846598, −13.47418698563643417103164099507, −12.49069880993942972358328532515, −10.15702947682389596226018116087, −7.39648249153949347970601731558, −6.56490519086549436063991939177, −4.85293607944907548520587009354, −2.49002079785047663088228635367, −1.28125965015893501421822210043, 1.86054716661326145017518195316, 3.17310513816787989147301863144, 5.20395802766106609942372557335, 5.86049243862730953523328929384, 9.395896551932272546319881022499, 10.31938973541374239176441039076, 12.23207718174622117104198207967, 13.98345999464700339526019691992, 15.14723070451998813843879406523, 16.64494894420700165525952218939

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