Properties

Label 2-2e2-4.3-c42-0-5
Degree $2$
Conductor $4$
Sign $-0.942 - 0.334i$
Analytic cond. $44.6910$
Root an. cond. $6.68513$
Motivic weight $42$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.06e6 + 3.55e5i)2-s + 9.12e9i·3-s + (4.14e12 + 1.46e12i)4-s − 1.85e14·5-s + (−3.24e15 + 1.88e16i)6-s + 7.39e17i·7-s + (8.04e18 + 4.51e18i)8-s + 2.61e19·9-s + (−3.84e20 − 6.60e19i)10-s − 7.81e20i·11-s + (−1.34e22 + 3.78e22i)12-s − 1.76e23·13-s + (−2.63e23 + 1.52e24i)14-s − 1.69e24i·15-s + (1.50e25 + 1.21e25i)16-s + 1.77e25·17-s + ⋯
L(s)  = 1  + (0.985 + 0.169i)2-s + 0.872i·3-s + (0.942 + 0.334i)4-s − 0.389·5-s + (−0.147 + 0.859i)6-s + 1.32i·7-s + (0.872 + 0.489i)8-s + 0.239·9-s + (−0.384 − 0.0660i)10-s − 0.105i·11-s + (−0.291 + 0.822i)12-s − 0.713·13-s + (−0.224 + 1.30i)14-s − 0.339i·15-s + (0.776 + 0.629i)16-s + 0.257·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.942 - 0.334i$
Analytic conductor: \(44.6910\)
Root analytic conductor: \(6.68513\)
Motivic weight: \(42\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :21),\ -0.942 - 0.334i)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(3.219568890\)
\(L(\frac12)\) \(\approx\) \(3.219568890\)
\(L(22)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.06e6 - 3.55e5i)T \)
good3 \( 1 - 9.12e9iT - 1.09e20T^{2} \)
5 \( 1 + 1.85e14T + 2.27e29T^{2} \)
7 \( 1 - 7.39e17iT - 3.11e35T^{2} \)
11 \( 1 + 7.81e20iT - 5.47e43T^{2} \)
13 \( 1 + 1.76e23T + 6.10e46T^{2} \)
17 \( 1 - 1.77e25T + 4.77e51T^{2} \)
19 \( 1 - 8.99e26iT - 5.10e53T^{2} \)
23 \( 1 + 2.55e28iT - 1.55e57T^{2} \)
29 \( 1 + 1.56e30T + 2.63e61T^{2} \)
31 \( 1 + 3.90e31iT - 4.33e62T^{2} \)
37 \( 1 - 2.18e32T + 7.31e65T^{2} \)
41 \( 1 + 1.09e34T + 5.45e67T^{2} \)
43 \( 1 - 3.18e34iT - 4.03e68T^{2} \)
47 \( 1 - 2.91e34iT - 1.69e70T^{2} \)
53 \( 1 + 2.53e36T + 2.62e72T^{2} \)
59 \( 1 + 2.40e36iT - 2.37e74T^{2} \)
61 \( 1 - 4.12e37T + 9.63e74T^{2} \)
67 \( 1 - 3.27e37iT - 4.95e76T^{2} \)
71 \( 1 + 5.97e38iT - 5.66e77T^{2} \)
73 \( 1 - 8.93e38T + 1.81e78T^{2} \)
79 \( 1 - 7.24e39iT - 5.01e79T^{2} \)
83 \( 1 - 1.04e40iT - 3.99e80T^{2} \)
89 \( 1 - 1.28e41T + 7.48e81T^{2} \)
97 \( 1 - 5.53e41T + 2.78e83T^{2} \)
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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

−15.47833868193069560074917151437, −14.66803450972796596306815673895, −12.66489057313141726330227714582, −11.53858162821743795966241675485, −9.814829697491281436922374423871, −7.921950394157079108967582818760, −6.00266586811925285748098261257, −4.77793476529342131588221462290, −3.54506070486837218452383444613, −2.10728881857780187029686475283, 0.59777837763148835093837636945, 1.82775407987288592398449177584, 3.52111905070989101773680258475, 4.84336972645396004996850674960, 6.83400859728766604029641575251, 7.49944175359751651210244423731, 10.27380684398293932141630718573, 11.77547875493695766279022229153, 13.05216056860468220372227861090, 14.01790158034436133037047780444

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