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Properties

Label 2-2e2-4.3-c24-0-8
Degree $2$
Conductor $4$
Sign $-0.863 + 0.503i$
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  + (−1.06e3 − 3.95e3i)2-s − 1.93e5i·3-s + (−1.44e7 + 8.44e6i)4-s + 3.74e8·5-s + (−7.66e8 + 2.07e8i)6-s − 2.47e10i·7-s + (4.88e10 + 4.82e10i)8-s + 2.44e11·9-s + (−3.99e11 − 1.47e12i)10-s + 7.20e11i·11-s + (1.63e12 + 2.81e12i)12-s − 1.13e13·13-s + (−9.79e13 + 2.64e13i)14-s − 7.25e13i·15-s + (1.38e14 − 2.44e14i)16-s − 7.24e14·17-s + ⋯
L(s)  = 1  + (−0.260 − 0.965i)2-s − 0.364i·3-s + (−0.863 + 0.503i)4-s + 1.53·5-s + (−0.352 + 0.0951i)6-s − 1.78i·7-s + (0.711 + 0.702i)8-s + 0.866·9-s + (−0.399 − 1.47i)10-s + 0.229i·11-s + (0.183 + 0.315i)12-s − 0.487·13-s + (−1.72 + 0.466i)14-s − 0.559i·15-s + (0.492 − 0.870i)16-s − 1.24·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.863 + 0.503i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.863 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.863 + 0.503i$
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: \((2,\ 4,\ (\ :12),\ -0.863 + 0.503i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.467739 - 1.73097i\)
\(L(\frac12)\) \(\approx\) \(0.467739 - 1.73097i\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06e3 + 3.95e3i)T \)
good3 \( 1 + 1.93e5iT - 2.82e11T^{2} \)
5 \( 1 - 3.74e8T + 5.96e16T^{2} \)
7 \( 1 + 2.47e10iT - 1.91e20T^{2} \)
11 \( 1 - 7.20e11iT - 9.84e24T^{2} \)
13 \( 1 + 1.13e13T + 5.42e26T^{2} \)
17 \( 1 + 7.24e14T + 3.39e29T^{2} \)
19 \( 1 + 1.99e15iT - 4.89e30T^{2} \)
23 \( 1 - 4.43e15iT - 4.80e32T^{2} \)
29 \( 1 - 2.02e16T + 1.25e35T^{2} \)
31 \( 1 + 4.90e17iT - 6.20e35T^{2} \)
37 \( 1 - 1.00e17T + 4.33e37T^{2} \)
41 \( 1 - 9.03e18T + 5.09e38T^{2} \)
43 \( 1 + 4.53e19iT - 1.59e39T^{2} \)
47 \( 1 - 1.32e20iT - 1.35e40T^{2} \)
53 \( 1 - 2.48e20T + 2.41e41T^{2} \)
59 \( 1 - 3.17e21iT - 3.16e42T^{2} \)
61 \( 1 - 2.83e21T + 7.04e42T^{2} \)
67 \( 1 + 8.09e21iT - 6.69e43T^{2} \)
71 \( 1 + 7.22e21iT - 2.69e44T^{2} \)
73 \( 1 - 2.43e21T + 5.24e44T^{2} \)
79 \( 1 - 8.02e22iT - 3.49e45T^{2} \)
83 \( 1 - 5.98e22iT - 1.14e46T^{2} \)
89 \( 1 - 1.56e23T + 6.10e46T^{2} \)
97 \( 1 - 2.43e23T + 4.81e47T^{2} \)
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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

−17.91871875665422754363970407635, −17.12088618845792765780681319697, −13.76051941851111614409429708159, −13.09376535272038979691314121663, −10.65628868110823304086007087311, −9.590258529709587723638336898139, −7.10007261726244055614304710398, −4.42657688355296903364093554124, −2.11605029434647920423339990168, −0.834497535102378126832878795640, 1.96214249419511736445029329185, 5.12345127493947565280807956498, 6.32253103690637790232243979147, 8.867308428519642790145452050813, 9.936183917095281794727227139297, 12.95660930057321467836526781142, 14.66684832897847493903844809635, 15.99433076007441132413824873679, 17.71132416410377599278548288699, 18.68237466774076985728551290023

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