Properties

Label 2-2e2-4.3-c42-0-11
Degree $2$
Conductor $4$
Sign $-0.133 - 0.990i$
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  + (1.57e6 + 1.38e6i)2-s + 7.51e9i·3-s + (5.89e11 + 4.35e12i)4-s + 5.33e14·5-s + (−1.03e16 + 1.18e16i)6-s − 7.46e17i·7-s + (−5.08e18 + 7.69e18i)8-s + 5.28e19·9-s + (8.42e20 + 7.35e20i)10-s + 6.22e21i·11-s + (−3.27e22 + 4.42e21i)12-s + 4.81e23·13-s + (1.03e24 − 1.17e24i)14-s + 4.00e24i·15-s + (−1.86e25 + 5.13e24i)16-s + 7.46e25·17-s + ⋯
L(s)  = 1  + (0.752 + 0.658i)2-s + 0.718i·3-s + (0.133 + 0.990i)4-s + 1.11·5-s + (−0.472 + 0.541i)6-s − 1.33i·7-s + (−0.551 + 0.834i)8-s + 0.483·9-s + (0.842 + 0.735i)10-s + 0.841i·11-s + (−0.712 + 0.0962i)12-s + 1.94·13-s + (0.879 − 1.00i)14-s + 0.803i·15-s + (−0.964 + 0.265i)16-s + 1.07·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 - 0.990i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (-0.133 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.133 - 0.990i$
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: \((2,\ 4,\ (\ :21),\ -0.133 - 0.990i)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(4.565388220\)
\(L(\frac12)\) \(\approx\) \(4.565388220\)
\(L(22)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.57e6 - 1.38e6i)T \)
good3 \( 1 - 7.51e9iT - 1.09e20T^{2} \)
5 \( 1 - 5.33e14T + 2.27e29T^{2} \)
7 \( 1 + 7.46e17iT - 3.11e35T^{2} \)
11 \( 1 - 6.22e21iT - 5.47e43T^{2} \)
13 \( 1 - 4.81e23T + 6.10e46T^{2} \)
17 \( 1 - 7.46e25T + 4.77e51T^{2} \)
19 \( 1 + 7.57e26iT - 5.10e53T^{2} \)
23 \( 1 - 2.53e28iT - 1.55e57T^{2} \)
29 \( 1 + 1.54e30T + 2.63e61T^{2} \)
31 \( 1 + 1.96e31iT - 4.33e62T^{2} \)
37 \( 1 + 6.73e32T + 7.31e65T^{2} \)
41 \( 1 - 3.14e33T + 5.45e67T^{2} \)
43 \( 1 - 7.71e33iT - 4.03e68T^{2} \)
47 \( 1 - 1.95e35iT - 1.69e70T^{2} \)
53 \( 1 + 3.91e35T + 2.62e72T^{2} \)
59 \( 1 - 2.85e36iT - 2.37e74T^{2} \)
61 \( 1 + 5.39e37T + 9.63e74T^{2} \)
67 \( 1 + 4.98e37iT - 4.95e76T^{2} \)
71 \( 1 + 1.04e39iT - 5.66e77T^{2} \)
73 \( 1 + 1.07e39T + 1.81e78T^{2} \)
79 \( 1 + 1.50e39iT - 5.01e79T^{2} \)
83 \( 1 - 3.31e40iT - 3.99e80T^{2} \)
89 \( 1 + 1.74e40T + 7.48e81T^{2} \)
97 \( 1 - 7.69e41T + 2.78e83T^{2} \)
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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

−15.50972453133940981673119743997, −13.92312564108696245245565586948, −13.11375129080830884044270198356, −10.82608934290985703973179511920, −9.461722120409867978081761307382, −7.42783671987452686303757321201, −6.03592106523003315561305991991, −4.56602667047614179933827331364, −3.50567632158542764502674848469, −1.42301990763378486909521183451, 1.17217715376840172253686565457, 1.94652957806138459651503770690, 3.39970537968706071229477584569, 5.63563551990945040059624867670, 6.20154085358784705127653154526, 8.742270472725275460563983969039, 10.34371054880764804274975072286, 12.01176855765646382728652860814, 13.14122624593905588953407526604, 14.15813890270161098920364295689

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