Properties

Label 2-2e2-4.3-c42-0-15
Degree $2$
Conductor $4$
Sign $-0.917 - 0.397i$
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  + (−2.05e6 − 4.25e5i)2-s + 8.66e9i·3-s + (4.03e12 + 1.74e12i)4-s − 5.87e14·5-s + (3.68e15 − 1.77e16i)6-s − 1.08e18i·7-s + (−7.54e18 − 5.30e18i)8-s + 3.43e19·9-s + (1.20e21 + 2.49e20i)10-s − 1.54e21i·11-s + (−1.51e22 + 3.49e22i)12-s − 2.11e23·13-s + (−4.60e23 + 2.22e24i)14-s − 5.09e24i·15-s + (1.32e25 + 1.40e25i)16-s + 9.80e25·17-s + ⋯
L(s)  = 1  + (−0.979 − 0.202i)2-s + 0.828i·3-s + (0.917 + 0.397i)4-s − 1.23·5-s + (0.167 − 0.811i)6-s − 1.94i·7-s + (−0.818 − 0.574i)8-s + 0.313·9-s + (1.20 + 0.249i)10-s − 0.209i·11-s + (−0.328 + 0.760i)12-s − 0.857·13-s + (−0.393 + 1.90i)14-s − 1.02i·15-s + (0.684 + 0.728i)16-s + 1.41·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (-0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.917 - 0.397i$
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: \((2,\ 4,\ (\ :21),\ -0.917 - 0.397i)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(0.002795545580\)
\(L(\frac12)\) \(\approx\) \(0.002795545580\)
\(L(22)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.05e6 + 4.25e5i)T \)
good3 \( 1 - 8.66e9iT - 1.09e20T^{2} \)
5 \( 1 + 5.87e14T + 2.27e29T^{2} \)
7 \( 1 + 1.08e18iT - 3.11e35T^{2} \)
11 \( 1 + 1.54e21iT - 5.47e43T^{2} \)
13 \( 1 + 2.11e23T + 6.10e46T^{2} \)
17 \( 1 - 9.80e25T + 4.77e51T^{2} \)
19 \( 1 - 6.85e26iT - 5.10e53T^{2} \)
23 \( 1 + 7.24e28iT - 1.55e57T^{2} \)
29 \( 1 - 5.49e29T + 2.63e61T^{2} \)
31 \( 1 + 3.73e30iT - 4.33e62T^{2} \)
37 \( 1 + 5.40e32T + 7.31e65T^{2} \)
41 \( 1 - 6.20e32T + 5.45e67T^{2} \)
43 \( 1 - 1.07e33iT - 4.03e68T^{2} \)
47 \( 1 + 2.86e34iT - 1.69e70T^{2} \)
53 \( 1 - 1.08e36T + 2.62e72T^{2} \)
59 \( 1 - 7.58e36iT - 2.37e74T^{2} \)
61 \( 1 + 2.45e37T + 9.63e74T^{2} \)
67 \( 1 + 2.23e38iT - 4.95e76T^{2} \)
71 \( 1 - 4.36e38iT - 5.66e77T^{2} \)
73 \( 1 + 1.36e39T + 1.81e78T^{2} \)
79 \( 1 - 5.82e38iT - 5.01e79T^{2} \)
83 \( 1 + 6.65e39iT - 3.99e80T^{2} \)
89 \( 1 + 3.51e40T + 7.48e81T^{2} \)
97 \( 1 + 6.45e41T + 2.78e83T^{2} \)
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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

−14.67527138596664400108356621777, −12.26158904844139131993645616892, −10.66948658598352264074176838503, −10.00674035166926400066702076595, −7.990467908329985157050451038919, −7.12407176774310141519099709601, −4.29713062374977908701930073495, −3.43249526437283645842338802487, −1.04071465763882415782031207998, −0.00130271441262928733071060248, 1.54994473268172913790997425568, 2.86352451578898157777156145118, 5.47930329791952300019854995912, 7.16441747453916469333809649407, 8.087077667091141318383073914455, 9.490428184744792080846382350594, 11.72173764430285417704850032900, 12.28158444054533487774203119650, 15.06426400361228417000359221584, 15.83224460396407859142905421598

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