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Properties

Label 2-28-28.23-c2-0-1
Degree $2$
Conductor $28$
Sign $0.543 - 0.839i$
Analytic cond. $0.762944$
Root an. cond. $0.873467$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.19 + 1.60i)2-s + (−1.86 + 1.07i)3-s + (−1.13 + 3.83i)4-s + (3.25 − 5.63i)5-s + (−3.96 − 1.70i)6-s + (2.39 − 6.57i)7-s + (−7.50 + 2.76i)8-s + (−2.17 + 3.76i)9-s + (12.9 − 1.52i)10-s + (−0.528 + 0.305i)11-s + (−2.01 − 8.38i)12-s − 10.6·13-s + (13.4 − 4.02i)14-s + 14.0i·15-s + (−13.4 − 8.71i)16-s + (5.99 + 10.3i)17-s + ⋯
L(s)  = 1  + (0.598 + 0.801i)2-s + (−0.622 + 0.359i)3-s + (−0.284 + 0.958i)4-s + (0.650 − 1.12i)5-s + (−0.660 − 0.283i)6-s + (0.342 − 0.939i)7-s + (−0.938 + 0.345i)8-s + (−0.241 + 0.418i)9-s + (1.29 − 0.152i)10-s + (−0.0480 + 0.0277i)11-s + (−0.167 − 0.699i)12-s − 0.820·13-s + (0.957 − 0.287i)14-s + 0.935i·15-s + (−0.838 − 0.544i)16-s + (0.352 + 0.610i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.543 - 0.839i$
Analytic conductor: \(0.762944\)
Root analytic conductor: \(0.873467\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :1),\ 0.543 - 0.839i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.961738 + 0.523283i\)
\(L(\frac12)\) \(\approx\) \(0.961738 + 0.523283i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 - 1.60i)T \)
7 \( 1 + (-2.39 + 6.57i)T \)
good3 \( 1 + (1.86 - 1.07i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-3.25 + 5.63i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (0.528 - 0.305i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 10.6T + 169T^{2} \)
17 \( 1 + (-5.99 - 10.3i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (10.5 + 6.10i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-34.7 - 20.0i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + 9.04T + 841T^{2} \)
31 \( 1 + (30.2 - 17.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-25.4 + 44.0i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 25.7T + 1.68e3T^{2} \)
43 \( 1 - 19.8iT - 1.84e3T^{2} \)
47 \( 1 + (-40.7 - 23.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-13.4 - 23.2i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (39.8 - 23.0i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (21.1 - 36.5i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-24.0 + 13.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 57.1iT - 5.04e3T^{2} \)
73 \( 1 + (28.1 + 48.7i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-15.9 - 9.19i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 37.3iT - 6.88e3T^{2} \)
89 \( 1 + (-12.3 + 21.3i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 109.T + 9.40e3T^{2} \)
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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

−17.02452717205798764278261494335, −16.34628304304116516830336268176, −14.76922620959158010665993993959, −13.51426697767290520631400632803, −12.58099028297368970940328524286, −10.92863986904341272502056368984, −9.156268756148422081714435909194, −7.53452713057067378281772367836, −5.59142114806708472563221911416, −4.58637172313302428525116826240, 2.64828301866322266445469852119, 5.40680002699864825858586071695, 6.62554897490320417107988604327, 9.337226413794549213702993290903, 10.75588145909543120886002167786, 11.77052236243200125098021501065, 12.83860537581552110621678902305, 14.46228199640345065256911722043, 14.99092087930229121921047168867, 17.20091412504267350249826152248

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