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Properties

Label 2-2e2-4.3-c12-0-4
Degree $2$
Conductor $4$
Sign $-0.978 + 0.205i$
Analytic cond. $3.65597$
Root an. cond. $1.91206$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−6.60 − 63.6i)2-s − 292. i·3-s + (−4.00e3 + 840. i)4-s − 1.53e4·5-s + (−1.86e4 + 1.93e3i)6-s − 1.29e5i·7-s + (7.99e4 + 2.49e5i)8-s + 4.45e5·9-s + (1.01e5 + 9.76e5i)10-s − 2.78e6i·11-s + (2.46e5 + 1.17e6i)12-s − 2.93e6·13-s + (−8.26e6 + 8.56e5i)14-s + 4.49e6i·15-s + (1.53e7 − 6.73e6i)16-s + 1.27e7·17-s + ⋯
L(s)  = 1  + (−0.103 − 0.994i)2-s − 0.401i·3-s + (−0.978 + 0.205i)4-s − 0.981·5-s + (−0.399 + 0.0414i)6-s − 1.10i·7-s + (0.305 + 0.952i)8-s + 0.838·9-s + (0.101 + 0.976i)10-s − 1.56i·11-s + (0.0824 + 0.393i)12-s − 0.607·13-s + (−1.09 + 0.113i)14-s + 0.394i·15-s + (0.915 − 0.401i)16-s + 0.530·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.978 + 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.978 + 0.205i$
Analytic conductor: \(3.65597\)
Root analytic conductor: \(1.91206\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :6),\ -0.978 + 0.205i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0940373 - 0.906932i\)
\(L(\frac12)\) \(\approx\) \(0.0940373 - 0.906932i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (6.60 + 63.6i)T \)
good3 \( 1 + 292. iT - 5.31e5T^{2} \)
5 \( 1 + 1.53e4T + 2.44e8T^{2} \)
7 \( 1 + 1.29e5iT - 1.38e10T^{2} \)
11 \( 1 + 2.78e6iT - 3.13e12T^{2} \)
13 \( 1 + 2.93e6T + 2.32e13T^{2} \)
17 \( 1 - 1.27e7T + 5.82e14T^{2} \)
19 \( 1 - 6.44e7iT - 2.21e15T^{2} \)
23 \( 1 + 1.17e8iT - 2.19e16T^{2} \)
29 \( 1 - 4.25e8T + 3.53e17T^{2} \)
31 \( 1 - 1.80e7iT - 7.87e17T^{2} \)
37 \( 1 + 1.18e9T + 6.58e18T^{2} \)
41 \( 1 - 6.12e9T + 2.25e19T^{2} \)
43 \( 1 + 4.83e9iT - 3.99e19T^{2} \)
47 \( 1 + 7.43e9iT - 1.16e20T^{2} \)
53 \( 1 - 9.24e9T + 4.91e20T^{2} \)
59 \( 1 - 1.33e10iT - 1.77e21T^{2} \)
61 \( 1 + 6.45e10T + 2.65e21T^{2} \)
67 \( 1 - 7.92e9iT - 8.18e21T^{2} \)
71 \( 1 + 1.35e11iT - 1.64e22T^{2} \)
73 \( 1 - 2.30e11T + 2.29e22T^{2} \)
79 \( 1 - 4.02e11iT - 5.90e22T^{2} \)
83 \( 1 + 5.54e10iT - 1.06e23T^{2} \)
89 \( 1 - 3.95e11T + 2.46e23T^{2} \)
97 \( 1 - 1.39e11T + 6.93e23T^{2} \)
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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

−21.22761702621877966734146653179, −19.64491407718745917667859880783, −18.68288423865416073995387643615, −16.59326636556613455665569238351, −13.91593105243877145839016333657, −12.21694767713942427770218362724, −10.50096852388341840474574081334, −7.896652903780465842880229192241, −3.84649909323247077742291957421, −0.72879330336756317193554392471, 4.66883199973894800937775704340, 7.40386975003873379608011877885, 9.510972574695669194449752822850, 12.43198164603991583016293699651, 15.09193550900033213370988424484, 15.74325440525839205848352435901, 17.81423330996556964236495733329, 19.38893837542276495514177958109, 21.72984944534935366053227064165, 23.13068473190794649570566134223

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