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Properties

Label 2-28-28.27-c5-0-2
Degree $2$
Conductor $28$
Sign $0.953 - 0.301i$
Analytic cond. $4.49074$
Root an. cond. $2.11913$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.36 − 3.60i)2-s − 28.2·3-s + (6.03 + 31.4i)4-s − 57.4i·5-s + (123. + 101. i)6-s + (15.0 + 128. i)7-s + (86.8 − 158. i)8-s + 552.·9-s + (−206. + 250. i)10-s − 158. i·11-s + (−170. − 886. i)12-s + 470. i·13-s + (398. − 615. i)14-s + 1.62e3i·15-s + (−951. + 379. i)16-s + 1.26e3i·17-s + ⋯
L(s)  = 1  + (−0.770 − 0.636i)2-s − 1.80·3-s + (0.188 + 0.982i)4-s − 1.02i·5-s + (1.39 + 1.15i)6-s + (0.115 + 0.993i)7-s + (0.479 − 0.877i)8-s + 2.27·9-s + (−0.654 + 0.792i)10-s − 0.396i·11-s + (−0.341 − 1.77i)12-s + 0.772i·13-s + (0.543 − 0.839i)14-s + 1.85i·15-s + (−0.928 + 0.370i)16-s + 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.953 - 0.301i$
Analytic conductor: \(4.49074\)
Root analytic conductor: \(2.11913\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :5/2),\ 0.953 - 0.301i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.478868 + 0.0738386i\)
\(L(\frac12)\) \(\approx\) \(0.478868 + 0.0738386i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.36 + 3.60i)T \)
7 \( 1 + (-15.0 - 128. i)T \)
good3 \( 1 + 28.2T + 243T^{2} \)
5 \( 1 + 57.4iT - 3.12e3T^{2} \)
11 \( 1 + 158. iT - 1.61e5T^{2} \)
13 \( 1 - 470. iT - 3.71e5T^{2} \)
17 \( 1 - 1.26e3iT - 1.41e6T^{2} \)
19 \( 1 - 789.T + 2.47e6T^{2} \)
23 \( 1 - 97.3iT - 6.43e6T^{2} \)
29 \( 1 - 3.57e3T + 2.05e7T^{2} \)
31 \( 1 - 7.66e3T + 2.86e7T^{2} \)
37 \( 1 + 6.34e3T + 6.93e7T^{2} \)
41 \( 1 - 1.61e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.21e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.28e4T + 2.29e8T^{2} \)
53 \( 1 - 1.19e4T + 4.18e8T^{2} \)
59 \( 1 - 1.31e4T + 7.14e8T^{2} \)
61 \( 1 - 2.13e3iT - 8.44e8T^{2} \)
67 \( 1 + 1.34e4iT - 1.35e9T^{2} \)
71 \( 1 - 7.74e3iT - 1.80e9T^{2} \)
73 \( 1 - 5.83e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.57e4iT - 3.07e9T^{2} \)
83 \( 1 - 5.27e4T + 3.93e9T^{2} \)
89 \( 1 + 9.63e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.18e5iT - 8.58e9T^{2} \)
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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

−16.57914312937675109321003735929, −15.81699927659788307992282503492, −12.94034232903404844236023951880, −12.06469607800209855484556710354, −11.35515617532005015872990957680, −9.910468239549724360490536413502, −8.464305667631667956649496685919, −6.33473232407988753414621831952, −4.74749196901925046651267272342, −1.20668582081529428990655144127, 0.61588871435668361057384036847, 5.06184295032681057121071819190, 6.59028911456178568627901777531, 7.37408698647723786904596359813, 10.12207801949499959528223002386, 10.65584311974618916823297767946, 11.82503089116494591127862736216, 13.83549826730643159284026909973, 15.40415011884580146206400687173, 16.37825322912859679292907092838

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