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Properties

Label 2-2e2-4.3-c24-0-4
Degree $2$
Conductor $4$
Sign $0.920 + 0.390i$
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  + (−4.01e3 − 815. i)2-s − 6.66e5i·3-s + (1.54e7 + 6.55e6i)4-s + 9.84e7·5-s + (−5.43e8 + 2.67e9i)6-s + 1.39e10i·7-s + (−5.66e10 − 3.88e10i)8-s − 1.61e11·9-s + (−3.95e11 − 8.03e10i)10-s + 4.84e12i·11-s + (4.36e12 − 1.02e13i)12-s + 2.94e13·13-s + (1.13e13 − 5.60e13i)14-s − 6.56e13i·15-s + (1.95e14 + 2.02e14i)16-s + 4.12e14·17-s + ⋯
L(s)  = 1  + (−0.979 − 0.199i)2-s − 1.25i·3-s + (0.920 + 0.390i)4-s + 0.403·5-s + (−0.249 + 1.22i)6-s + 1.00i·7-s + (−0.824 − 0.565i)8-s − 0.571·9-s + (−0.395 − 0.0803i)10-s + 1.54i·11-s + (0.489 − 1.15i)12-s + 1.26·13-s + (0.200 − 0.988i)14-s − 0.505i·15-s + (0.695 + 0.718i)16-s + 0.708·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.920 + 0.390i$
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: \((2,\ 4,\ (\ :12),\ 0.920 + 0.390i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(1.30201 - 0.264668i\)
\(L(\frac12)\) \(\approx\) \(1.30201 - 0.264668i\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.01e3 + 815. i)T \)
good3 \( 1 + 6.66e5iT - 2.82e11T^{2} \)
5 \( 1 - 9.84e7T + 5.96e16T^{2} \)
7 \( 1 - 1.39e10iT - 1.91e20T^{2} \)
11 \( 1 - 4.84e12iT - 9.84e24T^{2} \)
13 \( 1 - 2.94e13T + 5.42e26T^{2} \)
17 \( 1 - 4.12e14T + 3.39e29T^{2} \)
19 \( 1 + 2.77e15iT - 4.89e30T^{2} \)
23 \( 1 - 2.41e16iT - 4.80e32T^{2} \)
29 \( 1 - 2.94e17T + 1.25e35T^{2} \)
31 \( 1 + 3.19e17iT - 6.20e35T^{2} \)
37 \( 1 - 1.20e19T + 4.33e37T^{2} \)
41 \( 1 + 3.30e18T + 5.09e38T^{2} \)
43 \( 1 - 4.90e19iT - 1.59e39T^{2} \)
47 \( 1 - 9.94e18iT - 1.35e40T^{2} \)
53 \( 1 + 3.93e20T + 2.41e41T^{2} \)
59 \( 1 - 1.47e21iT - 3.16e42T^{2} \)
61 \( 1 - 1.47e21T + 7.04e42T^{2} \)
67 \( 1 + 5.87e21iT - 6.69e43T^{2} \)
71 \( 1 + 8.66e20iT - 2.69e44T^{2} \)
73 \( 1 - 2.68e22T + 5.24e44T^{2} \)
79 \( 1 + 6.58e22iT - 3.49e45T^{2} \)
83 \( 1 - 1.18e23iT - 1.14e46T^{2} \)
89 \( 1 - 4.38e22T + 6.10e46T^{2} \)
97 \( 1 - 2.57e23T + 4.81e47T^{2} \)
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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

−18.33233317774844734542517566447, −17.68435613886225417480790440824, −15.48967945026060993331279442014, −13.01564729726259565105812910775, −11.72419292285308581740221717831, −9.483329180503536680274124306735, −7.76282506033988633589492444436, −6.27020965563212024568455657415, −2.37114287227049341764265823183, −1.23425324513571243094582148926, 0.918563957108077587433765121413, 3.59359233130208033489994128394, 5.98968211017898841575407939129, 8.367486694815356415038126115912, 10.02513400940252838957098067348, 10.95817091563551131519033565302, 14.15227359911045417039620313791, 16.05539571098858253722275604656, 16.76197638288567311065692405228, 18.66601270563881390014420182415

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