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Properties

Label 2-2e2-4.3-c42-0-19
Degree $2$
Conductor $4$
Sign $0.975 + 0.220i$
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  + (2.32e5 − 2.08e6i)2-s − 2.00e10i·3-s + (−4.29e12 − 9.68e11i)4-s − 5.01e14·5-s + (−4.17e16 − 4.65e15i)6-s − 9.07e16i·7-s + (−3.01e18 + 8.71e18i)8-s − 2.92e20·9-s + (−1.16e20 + 1.04e21i)10-s − 1.06e22i·11-s + (−1.94e22 + 8.60e22i)12-s − 1.14e23·13-s + (−1.89e23 − 2.10e22i)14-s + 1.00e25i·15-s + (1.74e25 + 8.30e24i)16-s + 4.41e25·17-s + ⋯
L(s)  = 1  + (0.110 − 0.993i)2-s − 1.91i·3-s + (−0.975 − 0.220i)4-s − 1.05·5-s + (−1.90 − 0.212i)6-s − 0.162i·7-s + (−0.326 + 0.945i)8-s − 2.67·9-s + (−0.116 + 1.04i)10-s − 1.44i·11-s + (−0.422 + 1.86i)12-s − 0.461·13-s + (−0.161 − 0.0180i)14-s + 2.01i·15-s + (0.903 + 0.429i)16-s + 0.639·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.975 + 0.220i$
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: \((2,\ 4,\ (\ :21),\ 0.975 + 0.220i)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(0.5057357399\)
\(L(\frac12)\) \(\approx\) \(0.5057357399\)
\(L(22)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.32e5 + 2.08e6i)T \)
good3 \( 1 + 2.00e10iT - 1.09e20T^{2} \)
5 \( 1 + 5.01e14T + 2.27e29T^{2} \)
7 \( 1 + 9.07e16iT - 3.11e35T^{2} \)
11 \( 1 + 1.06e22iT - 5.47e43T^{2} \)
13 \( 1 + 1.14e23T + 6.10e46T^{2} \)
17 \( 1 - 4.41e25T + 4.77e51T^{2} \)
19 \( 1 + 9.90e25iT - 5.10e53T^{2} \)
23 \( 1 + 3.40e28iT - 1.55e57T^{2} \)
29 \( 1 + 5.31e30T + 2.63e61T^{2} \)
31 \( 1 + 3.57e31iT - 4.33e62T^{2} \)
37 \( 1 + 9.70e32T + 7.31e65T^{2} \)
41 \( 1 - 5.96e33T + 5.45e67T^{2} \)
43 \( 1 + 1.94e34iT - 4.03e68T^{2} \)
47 \( 1 - 2.30e34iT - 1.69e70T^{2} \)
53 \( 1 - 1.67e35T + 2.62e72T^{2} \)
59 \( 1 + 2.06e37iT - 2.37e74T^{2} \)
61 \( 1 - 2.37e37T + 9.63e74T^{2} \)
67 \( 1 - 4.39e37iT - 4.95e76T^{2} \)
71 \( 1 - 5.38e38iT - 5.66e77T^{2} \)
73 \( 1 + 1.46e39T + 1.81e78T^{2} \)
79 \( 1 - 1.90e39iT - 5.01e79T^{2} \)
83 \( 1 + 3.93e39iT - 3.99e80T^{2} \)
89 \( 1 - 4.67e40T + 7.48e81T^{2} \)
97 \( 1 - 8.04e41T + 2.78e83T^{2} \)
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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

−13.17375829296660393198971588110, −12.03879441366333334478349414388, −11.19011725526838831389814440443, −8.570898470120972310025111541261, −7.56045456897405612326006954708, −5.77311949009724690050335967050, −3.53917514183911786173005853644, −2.26844617464907632468194962804, −0.76662232502754801021509752852, −0.20290250020701537790035552082, 3.42602202458127475991370427065, 4.38856918901034930388385138571, 5.35483160238252149895463909909, 7.53857861507094066842493141585, 9.043150248094625882194727365389, 10.16651535052139395097703707978, 12.02497244866954472079598082871, 14.55864414270884928933388754358, 15.33007758170240984044194795458, 16.17281859081830022337060095032

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