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Properties

Label 2-28-28.23-c2-0-2
Degree $2$
Conductor $28$
Sign $0.867 + 0.498i$
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  + (−1.98 − 0.234i)2-s + (1.86 − 1.07i)3-s + (3.88 + 0.933i)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 + (−7.78 + 10.4i)10-s + (0.528 − 0.305i)11-s + (8.27 − 2.45i)12-s − 10.6·13-s + (6.30 − 12.4i)14-s − 14.0i·15-s + (14.2 + 7.26i)16-s + (5.99 + 10.3i)17-s + ⋯
L(s)  = 1  + (−0.993 − 0.117i)2-s + (0.622 − 0.359i)3-s + (0.972 + 0.233i)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 + (−0.778 + 1.04i)10-s + (0.0480 − 0.0277i)11-s + (0.689 − 0.204i)12-s − 0.820·13-s + (0.450 − 0.892i)14-s − 0.935i·15-s + (0.891 + 0.453i)16-s + (0.352 + 0.610i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.867 + 0.498i$
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: \((2,\ 28,\ (\ :1),\ 0.867 + 0.498i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.764569 - 0.204033i\)
\(L(\frac12)\) \(\approx\) \(0.764569 - 0.204033i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.98 + 0.234i)T \)
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

−16.95802586267642651270797252476, −16.10562390644000961370389959529, −14.55307961555312000590470449178, −12.94133172440954409781016499775, −11.99043030427738093964802436835, −9.972869237206570368766616457825, −8.921450163121076068139546603089, −7.929086645808156414107616025178, −5.79865542735556204358818929647, −2.22483127073442215302293163435, 2.98745174995511341857604813671, 6.41853903643882810314742754568, 7.68679038347286927706794573036, 9.612276170856846744413777413539, 10.11638158366932914336739878101, 11.64984676158272257712460089833, 13.90923005406859278808768052034, 14.71409959105649575891078766114, 15.98919366678139094532750520276, 17.36548912933614610411838851487

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