Properties

Label 2-2e2-4.3-c18-0-3
Degree $2$
Conductor $4$
Sign $0.386 - 0.922i$
Analytic cond. $8.21544$
Root an. cond. $2.86625$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (283. + 426. i)2-s − 1.44e4i·3-s + (−1.01e5 + 2.41e5i)4-s + 2.88e6·5-s + (6.15e6 − 4.09e6i)6-s + 4.40e7i·7-s + (−1.31e8 + 2.53e7i)8-s + 1.78e8·9-s + (8.16e8 + 1.22e9i)10-s + 2.10e9i·11-s + (3.49e9 + 1.46e9i)12-s + 3.93e9·13-s + (−1.87e10 + 1.24e10i)14-s − 4.15e10i·15-s + (−4.81e10 − 4.90e10i)16-s + 4.81e10·17-s + ⋯
L(s)  = 1  + (0.553 + 0.832i)2-s − 0.733i·3-s + (−0.386 + 0.922i)4-s + 1.47·5-s + (0.610 − 0.406i)6-s + 1.09i·7-s + (−0.982 + 0.188i)8-s + 0.461·9-s + (0.816 + 1.22i)10-s + 0.893i·11-s + (0.676 + 0.283i)12-s + 0.371·13-s + (−0.907 + 0.603i)14-s − 1.08i·15-s + (−0.700 − 0.713i)16-s + 0.406·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(8.21544\)
Root analytic conductor: \(2.86625\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :9),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(2.19389 + 1.45888i\)
\(L(\frac12)\) \(\approx\) \(2.19389 + 1.45888i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-283. - 426. i)T \)
good3 \( 1 + 1.44e4iT - 3.87e8T^{2} \)
5 \( 1 - 2.88e6T + 3.81e12T^{2} \)
7 \( 1 - 4.40e7iT - 1.62e15T^{2} \)
11 \( 1 - 2.10e9iT - 5.55e18T^{2} \)
13 \( 1 - 3.93e9T + 1.12e20T^{2} \)
17 \( 1 - 4.81e10T + 1.40e22T^{2} \)
19 \( 1 + 5.84e11iT - 1.04e23T^{2} \)
23 \( 1 - 1.93e11iT - 3.24e24T^{2} \)
29 \( 1 + 2.28e13T + 2.10e26T^{2} \)
31 \( 1 - 1.98e13iT - 6.99e26T^{2} \)
37 \( 1 + 1.40e14T + 1.68e28T^{2} \)
41 \( 1 - 1.71e14T + 1.07e29T^{2} \)
43 \( 1 + 7.88e14iT - 2.52e29T^{2} \)
47 \( 1 - 2.15e14iT - 1.25e30T^{2} \)
53 \( 1 - 1.84e15T + 1.08e31T^{2} \)
59 \( 1 + 4.68e15iT - 7.50e31T^{2} \)
61 \( 1 + 1.73e15T + 1.36e32T^{2} \)
67 \( 1 - 7.27e15iT - 7.40e32T^{2} \)
71 \( 1 + 4.97e16iT - 2.10e33T^{2} \)
73 \( 1 - 6.38e16T + 3.46e33T^{2} \)
79 \( 1 - 3.88e16iT - 1.43e34T^{2} \)
83 \( 1 - 1.69e17iT - 3.49e34T^{2} \)
89 \( 1 + 3.39e17T + 1.22e35T^{2} \)
97 \( 1 + 4.57e17T + 5.77e35T^{2} \)
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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

−21.17949838548580580371931875296, −18.37083552178871219402111133851, −17.46330072159238929720647023256, −15.36203871069999162269314204327, −13.62507664002814395329498036461, −12.51807768528802492967014801148, −9.204280055533304364560496092074, −6.89741428271496985419608590177, −5.39690661472126108130539790636, −2.15904068958930417915114736109, 1.42474597316221655091993845697, 3.78110008995410958588569972112, 5.75606751670442189677249946435, 9.695424040479923896214831089040, 10.65492394099069807981486056861, 13.20381660885070798868409104502, 14.34078031755523971984390073611, 16.66889021788276483550085640342, 18.57878965371262115144369148179, 20.69345842186049667236216001857

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