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Properties

Label 2-28-7.3-c4-0-1
Degree $2$
Conductor $28$
Sign $0.921 + 0.387i$
Analytic cond. $2.89435$
Root an. cond. $1.70128$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.35 − 0.784i)3-s + (38.3 − 22.1i)5-s + (42.3 + 24.5i)7-s + (−39.2 − 68.0i)9-s + (−11.7 + 20.3i)11-s + 136. i·13-s − 69.4·15-s + (−227. − 131. i)17-s + (−387. + 223. i)19-s + (−38.3 − 66.6i)21-s + (374. + 648. i)23-s + (667. − 1.15e3i)25-s + 250. i·27-s + 406.·29-s + (−584. − 337. i)31-s + ⋯
L(s)  = 1  + (−0.151 − 0.0871i)3-s + (1.53 − 0.885i)5-s + (0.865 + 0.501i)7-s + (−0.484 − 0.839i)9-s + (−0.0971 + 0.168i)11-s + 0.806i·13-s − 0.308·15-s + (−0.786 − 0.454i)17-s + (−1.07 + 0.619i)19-s + (−0.0869 − 0.151i)21-s + (0.708 + 1.22i)23-s + (1.06 − 1.84i)25-s + 0.343i·27-s + 0.483·29-s + (−0.607 − 0.350i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.921 + 0.387i$
Analytic conductor: \(2.89435\)
Root analytic conductor: \(1.70128\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :2),\ 0.921 + 0.387i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.55340 - 0.313612i\)
\(L(\frac12)\) \(\approx\) \(1.55340 - 0.313612i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-42.3 - 24.5i)T \)
good3 \( 1 + (1.35 + 0.784i)T + (40.5 + 70.1i)T^{2} \)
5 \( 1 + (-38.3 + 22.1i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (11.7 - 20.3i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 - 136. iT - 2.85e4T^{2} \)
17 \( 1 + (227. + 131. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (387. - 223. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (-374. - 648. i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 - 406.T + 7.07e5T^{2} \)
31 \( 1 + (584. + 337. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (372. + 645. i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 - 2.47e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.63e3T + 3.41e6T^{2} \)
47 \( 1 + (579. - 334. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-1.01e3 + 1.75e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (1.01e3 + 585. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.02e3 - 1.16e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.77e3 + 6.54e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 7.57e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.28e3 - 1.89e3i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (3.89e3 + 6.75e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 2.18e3iT - 4.74e7T^{2} \)
89 \( 1 + (-8.05e3 + 4.64e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 + 9.98e3iT - 8.85e7T^{2} \)
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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

−16.74446766843733654455122525384, −15.00292285037885964820613862467, −13.86697844049336044157045620747, −12.68642173916509162703112111361, −11.38606255032621461100265094830, −9.558582809921076026651889991812, −8.674760391305448618824857250066, −6.30597490567660162390187619599, −5.00667049790379191886197397047, −1.80689940933409746069633287096, 2.31532916365648205235693675108, 5.16750799135441965390962438282, 6.67792386087337301726429095801, 8.540182932206538752322731057512, 10.49092140934562258115353080819, 10.85936506646538165828397582482, 13.14126412476617304548406991666, 14.04691261900454492100049185574, 15.03647174950760246300832044207, 16.99277902597652870338454631373

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