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Properties

Label 2-28-28.27-c3-0-1
Degree $2$
Conductor $28$
Sign $-0.906 - 0.422i$
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  + (0.414 + 2.79i)2-s − 7.54·3-s + (−7.65 + 2.31i)4-s + 8.74i·5-s + (−3.12 − 21.1i)6-s + (13.8 + 12.3i)7-s + (−9.65 − 20.4i)8-s + 29.9·9-s + (−24.4 + 3.62i)10-s + 0.397i·11-s + (57.7 − 17.4i)12-s + 75.7i·13-s + (−28.8 + 43.7i)14-s − 66.0i·15-s + (53.2 − 35.4i)16-s − 84.4i·17-s + ⋯
L(s)  = 1  + (0.146 + 0.989i)2-s − 1.45·3-s + (−0.957 + 0.289i)4-s + 0.782i·5-s + (−0.212 − 1.43i)6-s + (0.745 + 0.666i)7-s + (−0.426 − 0.904i)8-s + 1.11·9-s + (−0.773 + 0.114i)10-s + 0.0109i·11-s + (1.39 − 0.420i)12-s + 1.61i·13-s + (−0.550 + 0.834i)14-s − 1.13i·15-s + (0.832 − 0.554i)16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $-0.906 - 0.422i$
Analytic conductor: \(1.65205\)
Root analytic conductor: \(1.28532\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :3/2),\ -0.906 - 0.422i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.150115 + 0.677573i\)
\(L(\frac12)\) \(\approx\) \(0.150115 + 0.677573i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.414 - 2.79i)T \)
7 \( 1 + (-13.8 - 12.3i)T \)
good3 \( 1 + 7.54T + 27T^{2} \)
5 \( 1 - 8.74iT - 125T^{2} \)
11 \( 1 - 0.397iT - 1.33e3T^{2} \)
13 \( 1 - 75.7iT - 2.19e3T^{2} \)
17 \( 1 + 84.4iT - 4.91e3T^{2} \)
19 \( 1 + 20.0T + 6.85e3T^{2} \)
23 \( 1 - 122. iT - 1.21e4T^{2} \)
29 \( 1 + 175.T + 2.43e4T^{2} \)
31 \( 1 - 128.T + 2.97e4T^{2} \)
37 \( 1 - 253.T + 5.06e4T^{2} \)
41 \( 1 + 81.4iT - 6.89e4T^{2} \)
43 \( 1 - 69.1iT - 7.95e4T^{2} \)
47 \( 1 - 147.T + 1.03e5T^{2} \)
53 \( 1 - 283.T + 1.48e5T^{2} \)
59 \( 1 + 632.T + 2.05e5T^{2} \)
61 \( 1 - 3.25iT - 2.26e5T^{2} \)
67 \( 1 + 551. iT - 3.00e5T^{2} \)
71 \( 1 + 486. iT - 3.57e5T^{2} \)
73 \( 1 - 165. iT - 3.89e5T^{2} \)
79 \( 1 + 279. iT - 4.93e5T^{2} \)
83 \( 1 - 622.T + 5.71e5T^{2} \)
89 \( 1 + 696. iT - 7.04e5T^{2} \)
97 \( 1 - 1.62e3iT - 9.12e5T^{2} \)
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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

−17.14313836201934457527765192108, −16.20887855155579611997017751039, −14.99895922444871968710142088625, −13.83928267701633836865341779416, −12.02975876661250917136211742846, −11.17968577883811534207344534383, −9.309210760998392092766959099600, −7.27980167186517558903525740708, −6.10346089861409948963877345961, −4.78039936897250803215956837418, 0.827496651368178914574803496752, 4.49884344058728289473747602171, 5.70324632900367093919192695814, 8.281881905589440461482162223610, 10.33268201438454888314998771769, 11.02752368573687667813228936611, 12.35285319892081432358670507460, 13.09708586402415183294067291010, 14.89790480520606450050228104420, 16.78682827032431296996772346533

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