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Properties

Label 2-28-28.23-c4-0-4
Degree $2$
Conductor $28$
Sign $-0.718 - 0.695i$
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  + (1.03 + 3.86i)2-s + (−1.18 + 0.683i)3-s + (−13.8 + 7.99i)4-s + (−16.2 + 28.1i)5-s + (−3.86 − 3.86i)6-s + (47.8 − 10.4i)7-s + (−45.2 − 45.2i)8-s + (−39.5 + 68.5i)9-s + (−125. − 33.7i)10-s + (46.9 − 27.1i)11-s + (10.9 − 18.9i)12-s + 223.·13-s + (89.8 + 174. i)14-s − 44.4i·15-s + (128. − 221. i)16-s + (22.6 + 39.1i)17-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.131 + 0.0759i)3-s + (−0.866 + 0.499i)4-s + (−0.650 + 1.12i)5-s + (−0.107 − 0.107i)6-s + (0.977 − 0.213i)7-s + (−0.706 − 0.707i)8-s + (−0.488 + 0.846i)9-s + (−1.25 − 0.337i)10-s + (0.388 − 0.224i)11-s + (0.0759 − 0.131i)12-s + 1.32·13-s + (0.458 + 0.888i)14-s − 0.197i·15-s + (0.500 − 0.865i)16-s + (0.0782 + 0.135i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $-0.718 - 0.695i$
Analytic conductor: \(2.89435\)
Root analytic conductor: \(1.70128\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :2),\ -0.718 - 0.695i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.475293 + 1.17377i\)
\(L(\frac12)\) \(\approx\) \(0.475293 + 1.17377i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.03 - 3.86i)T \)
7 \( 1 + (-47.8 + 10.4i)T \)
good3 \( 1 + (1.18 - 0.683i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (16.2 - 28.1i)T + (-312.5 - 541. i)T^{2} \)
11 \( 1 + (-46.9 + 27.1i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 - 223.T + 2.85e4T^{2} \)
17 \( 1 + (-22.6 - 39.1i)T + (-4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-454. - 262. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (640. + 369. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 392.T + 7.07e5T^{2} \)
31 \( 1 + (494. - 285. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-162. + 281. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 110.T + 2.82e6T^{2} \)
43 \( 1 + 2.81e3iT - 3.41e6T^{2} \)
47 \( 1 + (-877. - 506. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-2.25e3 - 3.91e3i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-3.88e3 + 2.24e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (593. - 1.02e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.46e3 + 1.99e3i)T + (1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 + 2.62e3iT - 2.54e7T^{2} \)
73 \( 1 + (254. + 440. i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (8.41e3 + 4.85e3i)T + (1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 7.48e3iT - 4.74e7T^{2} \)
89 \( 1 + (-2.12e3 + 3.68e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 1.46e4T + 8.85e7T^{2} \)
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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

−16.68780267760308978159540476949, −15.68394671856760574199058632483, −14.40614702169899765219310762810, −13.84187411074542671267844440172, −11.76931870932301300508859771204, −10.65733039557870036413904452088, −8.418434733376100291340500128380, −7.40727910069569561267254728000, −5.74735519923544617956442717348, −3.81079373562539591273600297221, 1.08204620260190674660965471130, 3.93522899306839967085984844302, 5.48287639683392677973300248135, 8.302270820021118393258812745303, 9.331282827185544967790229144096, 11.43807866269242762629347874608, 11.86079646192840736822131178396, 13.24443453523348587867593060907, 14.55769269101559188169415124561, 15.92572899224841232059475521824

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