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Properties

Label 2-2e2-4.3-c24-0-9
Degree $2$
Conductor $4$
Sign $-0.359 + 0.933i$
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  + (2.31e3 + 3.37e3i)2-s − 9.62e5i·3-s + (−6.02e6 + 1.56e7i)4-s + 1.24e8·5-s + (3.25e9 − 2.23e9i)6-s − 3.87e9i·7-s + (−6.68e10 + 1.59e10i)8-s − 6.44e11·9-s + (2.88e11 + 4.19e11i)10-s − 5.11e12i·11-s + (1.50e13 + 5.80e12i)12-s − 2.88e13·13-s + (1.30e13 − 8.98e12i)14-s − 1.19e14i·15-s + (−2.08e14 − 1.88e14i)16-s + 2.36e14·17-s + ⋯
L(s)  = 1  + (0.566 + 0.824i)2-s − 1.81i·3-s + (−0.359 + 0.933i)4-s + 0.509·5-s + (1.49 − 1.02i)6-s − 0.279i·7-s + (−0.972 + 0.232i)8-s − 2.28·9-s + (0.288 + 0.419i)10-s − 1.62i·11-s + (1.69 + 0.650i)12-s − 1.24·13-s + (0.230 − 0.158i)14-s − 0.922i·15-s + (−0.742 − 0.670i)16-s + 0.406·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.359 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.359 + 0.933i$
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: \((2,\ 4,\ (\ :12),\ -0.359 + 0.933i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.957893 - 1.39501i\)
\(L(\frac12)\) \(\approx\) \(0.957893 - 1.39501i\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.31e3 - 3.37e3i)T \)
good3 \( 1 + 9.62e5iT - 2.82e11T^{2} \)
5 \( 1 - 1.24e8T + 5.96e16T^{2} \)
7 \( 1 + 3.87e9iT - 1.91e20T^{2} \)
11 \( 1 + 5.11e12iT - 9.84e24T^{2} \)
13 \( 1 + 2.88e13T + 5.42e26T^{2} \)
17 \( 1 - 2.36e14T + 3.39e29T^{2} \)
19 \( 1 + 1.11e15iT - 4.89e30T^{2} \)
23 \( 1 + 9.54e15iT - 4.80e32T^{2} \)
29 \( 1 - 4.61e17T + 1.25e35T^{2} \)
31 \( 1 - 3.35e17iT - 6.20e35T^{2} \)
37 \( 1 - 7.73e17T + 4.33e37T^{2} \)
41 \( 1 - 5.81e18T + 5.09e38T^{2} \)
43 \( 1 - 2.40e19iT - 1.59e39T^{2} \)
47 \( 1 + 1.39e20iT - 1.35e40T^{2} \)
53 \( 1 - 6.00e20T + 2.41e41T^{2} \)
59 \( 1 - 1.87e20iT - 3.16e42T^{2} \)
61 \( 1 - 2.52e21T + 7.04e42T^{2} \)
67 \( 1 - 7.53e21iT - 6.69e43T^{2} \)
71 \( 1 + 1.51e22iT - 2.69e44T^{2} \)
73 \( 1 + 4.83e21T + 5.24e44T^{2} \)
79 \( 1 + 2.60e22iT - 3.49e45T^{2} \)
83 \( 1 - 1.50e22iT - 1.14e46T^{2} \)
89 \( 1 + 3.62e23T + 6.10e46T^{2} \)
97 \( 1 - 2.71e23T + 4.81e47T^{2} \)
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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

−17.94783762443741857946661087551, −16.79689193761006683640314801156, −14.21405884639900677602614235458, −13.36245038677964763321214675158, −11.96579276897534739258256506905, −8.363870963784358633743983050266, −6.97519768424119144103488184598, −5.70562525773032876064202740275, −2.69644239970067479368353127460, −0.55153169317243363075981047306, 2.44729407344652149033615218598, 4.23280228050894004430887988579, 5.36608163981823162885724962328, 9.548847562429216881321705844182, 10.18383264393607554616353923567, 12.07083814373153454882392671481, 14.41155731845809270408006247307, 15.41736684426091123681052210612, 17.42865383348580544180805197462, 19.87917314650419205606810494830

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