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Properties

Label 2-28-4.3-c4-0-4
Degree $2$
Conductor $28$
Sign $0.994 - 0.100i$
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  + (−2.68 + 2.96i)2-s − 5.27i·3-s + (−1.60 − 15.9i)4-s + 36.6·5-s + (15.6 + 14.1i)6-s + 18.5i·7-s + (51.5 + 37.9i)8-s + 53.1·9-s + (−98.1 + 108. i)10-s − 162. i·11-s + (−83.9 + 8.46i)12-s + 134.·13-s + (−54.9 − 49.6i)14-s − 193. i·15-s + (−250. + 51.1i)16-s − 386.·17-s + ⋯
L(s)  = 1  + (−0.670 + 0.741i)2-s − 0.585i·3-s + (−0.100 − 0.994i)4-s + 1.46·5-s + (0.434 + 0.392i)6-s + 0.377i·7-s + (0.805 + 0.592i)8-s + 0.656·9-s + (−0.981 + 1.08i)10-s − 1.34i·11-s + (−0.582 + 0.0587i)12-s + 0.796·13-s + (−0.280 − 0.253i)14-s − 0.857i·15-s + (−0.979 + 0.199i)16-s − 1.33·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.994 - 0.100i$
Analytic conductor: \(2.89435\)
Root analytic conductor: \(1.70128\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :2),\ 0.994 - 0.100i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.22039 + 0.0613771i\)
\(L(\frac12)\) \(\approx\) \(1.22039 + 0.0613771i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.68 - 2.96i)T \)
7 \( 1 - 18.5iT \)
good3 \( 1 + 5.27iT - 81T^{2} \)
5 \( 1 - 36.6T + 625T^{2} \)
11 \( 1 + 162. iT - 1.46e4T^{2} \)
13 \( 1 - 134.T + 2.85e4T^{2} \)
17 \( 1 + 386.T + 8.35e4T^{2} \)
19 \( 1 - 137. iT - 1.30e5T^{2} \)
23 \( 1 - 649. iT - 2.79e5T^{2} \)
29 \( 1 + 508.T + 7.07e5T^{2} \)
31 \( 1 - 573. iT - 9.23e5T^{2} \)
37 \( 1 + 237.T + 1.87e6T^{2} \)
41 \( 1 + 1.61e3T + 2.82e6T^{2} \)
43 \( 1 - 2.89e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.48e3iT - 4.87e6T^{2} \)
53 \( 1 - 823.T + 7.89e6T^{2} \)
59 \( 1 + 1.79e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.83e3T + 1.38e7T^{2} \)
67 \( 1 - 6.13e3iT - 2.01e7T^{2} \)
71 \( 1 + 7.65e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.65e3T + 2.83e7T^{2} \)
79 \( 1 - 4.28e3iT - 3.89e7T^{2} \)
83 \( 1 - 8.05e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.33e4T + 6.27e7T^{2} \)
97 \( 1 + 7.29e3T + 8.85e7T^{2} \)
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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

−16.63335056447547611522378681115, −15.47401290273283601364454613267, −13.81606564450418536766209878841, −13.30382058206888544635994226570, −11.03441301859835827380454754984, −9.638937258272885485124879466312, −8.477814440255929117099412373825, −6.64891706181051002963381151730, −5.66711976393855265560290527353, −1.60309698441057573753169644362, 1.95645905357875332991977778119, 4.43340926901935595211553114657, 6.88012106817417080672163632066, 9.024241462056267830672942322187, 9.968227250078105582618431957178, 10.80330373260965971769399324467, 12.72936002226860126331572348935, 13.61410421328497502826326718606, 15.41301372944403640764605771728, 16.81995387346026254730475711032

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