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Properties

Label 2-28-28.27-c3-0-7
Degree $2$
Conductor $28$
Sign $0.519 + 0.854i$
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 + (−40.2 + 33.4i)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 + (−0.768 + 0.639i)14-s + 1.13i·15-s + (0.832 + 0.554i)16-s − 1.20i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.519 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.519 + 0.854i$
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: \((2,\ 28,\ (\ :3/2),\ 0.519 + 0.854i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.38405 - 0.777783i\)
\(L(\frac12)\) \(\approx\) \(1.38405 - 0.777783i\)
\(L(\frac{5}{2})\) 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

−16.47383282798894204041250240481, −14.64459495515956807696259338016, −14.05974315417803607702895141650, −13.07323575463757044717729779073, −11.34669792071930662343835523551, −9.877095483545193107445618919174, −8.969454871118781243595045894826, −7.10163778777279024537604635861, −3.95081079768135551061333064583, −2.58636021178217177967268845173, 3.45690684757025700157866683486, 5.64626231607538918354913165006, 7.75758211372430954422725047858, 8.711849182723099400107826017529, 9.694823177388549913374401097782, 12.78812768786480770292740620316, 13.21154707375921978932232711072, 14.82851790120554633924199221269, 15.40016316323563811221355443666, 16.62690722285209789033340136085

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