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Properties

Label 2-28-28.19-c3-0-2
Degree $2$
Conductor $28$
Sign $0.937 - 0.348i$
Analytic cond. $1.65205$
Root an. cond. $1.28532$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.49 − 2.40i)2-s + (4.65 + 8.06i)3-s + (−3.54 + 7.16i)4-s + (5.11 + 2.95i)5-s + (12.4 − 23.2i)6-s + (1.92 − 18.4i)7-s + (22.5 − 2.16i)8-s + (−29.8 + 51.6i)9-s + (−0.533 − 16.6i)10-s + (0.267 − 0.154i)11-s + (−74.3 + 4.75i)12-s − 43.7i·13-s + (−47.1 + 22.8i)14-s + 54.9i·15-s + (−38.8 − 50.8i)16-s + (−27.0 + 15.6i)17-s + ⋯
L(s)  = 1  + (−0.527 − 0.849i)2-s + (0.895 + 1.55i)3-s + (−0.443 + 0.896i)4-s + (0.457 + 0.264i)5-s + (0.845 − 1.57i)6-s + (0.104 − 0.994i)7-s + (0.995 − 0.0957i)8-s + (−1.10 + 1.91i)9-s + (−0.0168 − 0.528i)10-s + (0.00733 − 0.00423i)11-s + (−1.78 + 0.114i)12-s − 0.932i·13-s + (−0.899 + 0.436i)14-s + 0.946i·15-s + (−0.606 − 0.795i)16-s + (−0.386 + 0.223i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.937 - 0.348i$
Analytic conductor: \(1.65205\)
Root analytic conductor: \(1.28532\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :3/2),\ 0.937 - 0.348i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.16747 + 0.210295i\)
\(L(\frac12)\) \(\approx\) \(1.16747 + 0.210295i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.49 + 2.40i)T \)
7 \( 1 + (-1.92 + 18.4i)T \)
good3 \( 1 + (-4.65 - 8.06i)T + (-13.5 + 23.3i)T^{2} \)
5 \( 1 + (-5.11 - 2.95i)T + (62.5 + 108. i)T^{2} \)
11 \( 1 + (-0.267 + 0.154i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 43.7iT - 2.19e3T^{2} \)
17 \( 1 + (27.0 - 15.6i)T + (2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-39.7 + 68.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-16.5 - 9.52i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 40.1T + 2.43e4T^{2} \)
31 \( 1 + (-42.8 - 74.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-72.4 + 125. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 254. iT - 6.89e4T^{2} \)
43 \( 1 - 366. iT - 7.95e4T^{2} \)
47 \( 1 + (5.12 - 8.87i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-315. - 546. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (257. + 446. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (669. + 386. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (450. - 259. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 261. iT - 3.57e5T^{2} \)
73 \( 1 + (-588. + 339. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (299. + 172. i)T + (2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 805.T + 5.71e5T^{2} \)
89 \( 1 + (40.7 + 23.5i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 763. iT - 9.12e5T^{2} \)
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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

−16.86030542831923439594164351650, −15.67130281095808996335305879607, −14.26307259970466111769431101296, −13.28502955708446515945157017154, −11.02615011502179764057834159661, −10.25025952938918426094034559446, −9.310817461839901129338068122541, −7.914949599242828351069300705835, −4.51104217868518588117348923760, −3.03172723846557677740783996652, 1.84803286112979944436215733835, 5.87496551643793317580591025453, 7.19826754142350639093327660683, 8.505329504612691334170171960371, 9.365305059021607187683573784276, 11.92137027181062949954406727758, 13.31963948701683024973550712031, 14.16899957519633373410932761819, 15.27421627409124043202434934339, 16.91553226582937517823234732702

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