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Properties

Label 2-2e2-4.3-c24-0-5
Degree $2$
Conductor $4$
Sign $-0.247 + 0.968i$
Analytic cond. $14.5986$
Root an. cond. $3.82082$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.51e3 − 3.23e3i)2-s + 3.86e5i·3-s + (−4.15e6 + 1.62e7i)4-s − 3.41e8·5-s + (1.25e9 − 9.70e8i)6-s + 1.51e10i·7-s + (6.30e10 − 2.73e10i)8-s + 1.33e11·9-s + (8.58e11 + 1.10e12i)10-s − 3.05e12i·11-s + (−6.28e12 − 1.60e12i)12-s − 2.98e13·13-s + (4.91e13 − 3.81e13i)14-s − 1.32e14i·15-s + (−2.46e14 − 1.35e14i)16-s + 3.86e14·17-s + ⋯
L(s)  = 1  + (−0.613 − 0.789i)2-s + 0.727i·3-s + (−0.247 + 0.968i)4-s − 1.40·5-s + (0.574 − 0.445i)6-s + 1.09i·7-s + (0.917 − 0.398i)8-s + 0.471·9-s + (0.858 + 1.10i)10-s − 0.971i·11-s + (−0.704 − 0.180i)12-s − 1.27·13-s + (0.866 − 0.672i)14-s − 1.01i·15-s + (−0.877 − 0.480i)16-s + 0.663·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.247 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.247 + 0.968i$
Analytic conductor: \(14.5986\)
Root analytic conductor: \(3.82082\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :12),\ -0.247 + 0.968i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.311089 - 0.400730i\)
\(L(\frac12)\) \(\approx\) \(0.311089 - 0.400730i\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.51e3 + 3.23e3i)T \)
good3 \( 1 - 3.86e5iT - 2.82e11T^{2} \)
5 \( 1 + 3.41e8T + 5.96e16T^{2} \)
7 \( 1 - 1.51e10iT - 1.91e20T^{2} \)
11 \( 1 + 3.05e12iT - 9.84e24T^{2} \)
13 \( 1 + 2.98e13T + 5.42e26T^{2} \)
17 \( 1 - 3.86e14T + 3.39e29T^{2} \)
19 \( 1 + 2.13e15iT - 4.89e30T^{2} \)
23 \( 1 + 1.14e16iT - 4.80e32T^{2} \)
29 \( 1 - 1.85e17T + 1.25e35T^{2} \)
31 \( 1 + 1.46e18iT - 6.20e35T^{2} \)
37 \( 1 - 3.27e18T + 4.33e37T^{2} \)
41 \( 1 - 2.93e19T + 5.09e38T^{2} \)
43 \( 1 - 3.99e19iT - 1.59e39T^{2} \)
47 \( 1 + 7.73e19iT - 1.35e40T^{2} \)
53 \( 1 + 5.81e20T + 2.41e41T^{2} \)
59 \( 1 + 2.78e20iT - 3.16e42T^{2} \)
61 \( 1 + 3.59e21T + 7.04e42T^{2} \)
67 \( 1 + 1.33e22iT - 6.69e43T^{2} \)
71 \( 1 + 2.03e22iT - 2.69e44T^{2} \)
73 \( 1 + 2.21e22T + 5.24e44T^{2} \)
79 \( 1 - 6.22e22iT - 3.49e45T^{2} \)
83 \( 1 - 3.74e22iT - 1.14e46T^{2} \)
89 \( 1 + 2.83e23T + 6.10e46T^{2} \)
97 \( 1 - 2.96e23T + 4.81e47T^{2} \)
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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

−18.66339403498853612056963796382, −16.48171741449453160934903044868, −15.26155326522770175700579407756, −12.39259574294762490853628267414, −11.20116723078328087769200097346, −9.396400777311429515866669079836, −7.84177316065088678902186208658, −4.45492195267057643694696303718, −2.88943589840874789155650490069, −0.30211461224610157728769112829, 1.17918331452048432111698713399, 4.40886659059171445985877298995, 7.18632043298122156630074793123, 7.68539876942215802440071871898, 10.16240157507341801607705703833, 12.33388269037199691072845898185, 14.44503585264416544721393097084, 15.97673963069368726911839189900, 17.48669400240809212829855469031, 19.09060064946029257116182485017

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