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Properties

Label 2-28-28.23-c2-0-0
Degree $2$
Conductor $28$
Sign $-0.788 - 0.615i$
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  + (−1.51 + 1.30i)2-s + (−3.95 + 2.28i)3-s + (0.597 − 3.95i)4-s + (−2.62 + 4.54i)5-s + (3.01 − 8.60i)6-s + (5.86 + 3.81i)7-s + (4.25 + 6.77i)8-s + (5.90 − 10.2i)9-s + (−1.95 − 10.3i)10-s + (−1.91 + 1.10i)11-s + (6.66 + 16.9i)12-s − 3.29·13-s + (−13.8 + 1.86i)14-s − 23.9i·15-s + (−15.2 − 4.72i)16-s + (6.69 + 11.6i)17-s + ⋯
L(s)  = 1  + (−0.758 + 0.652i)2-s + (−1.31 + 0.760i)3-s + (0.149 − 0.988i)4-s + (−0.525 + 0.909i)5-s + (0.502 − 1.43i)6-s + (0.838 + 0.545i)7-s + (0.531 + 0.846i)8-s + (0.655 − 1.13i)9-s + (−0.195 − 1.03i)10-s + (−0.174 + 0.100i)11-s + (0.555 + 1.41i)12-s − 0.253·13-s + (−0.991 + 0.133i)14-s − 1.59i·15-s + (−0.955 − 0.295i)16-s + (0.394 + 0.682i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $-0.788 - 0.615i$
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: \((2,\ 28,\ (\ :1),\ -0.788 - 0.615i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.139811 + 0.406058i\)
\(L(\frac12)\) \(\approx\) \(0.139811 + 0.406058i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.51 - 1.30i)T \)
7 \( 1 + (-5.86 - 3.81i)T \)
good3 \( 1 + (3.95 - 2.28i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (2.62 - 4.54i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (1.91 - 1.10i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 3.29T + 169T^{2} \)
17 \( 1 + (-6.69 - 11.6i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + (-6.72 - 3.88i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (3.66 + 2.11i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 - 39.4T + 841T^{2} \)
31 \( 1 + (-17.2 + 9.93i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-12.8 + 22.2i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 55.0T + 1.68e3T^{2} \)
43 \( 1 - 78.2iT - 1.84e3T^{2} \)
47 \( 1 + (18.3 + 10.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (24.0 + 41.6i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-66.7 + 38.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.7 + 41.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (45.2 - 26.1i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 90.1iT - 5.04e3T^{2} \)
73 \( 1 + (20.7 + 36.0i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-114. - 65.8i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 11.8iT - 6.88e3T^{2} \)
89 \( 1 + (15.5 - 26.9i)T + (-3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 140.T + 9.40e3T^{2} \)
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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

−17.50565165041620206666570133636, −16.34097772387881447088524105113, −15.35291532554817057308017749832, −14.53497906408402646658584135903, −11.82371267456278878291097249139, −10.95792966057093304199434551322, −9.972570562672271793678636438203, −8.026770635869523338248039385683, −6.37590751721768189499145050713, −4.97887499028383937925241511668, 0.902243226011562403949799107448, 4.84418028513637030504844735767, 7.14531919164706982098264221370, 8.375363918645870552016169936007, 10.36926461826523943070170753603, 11.69409643310490558642361843084, 12.14703243840487657281044677744, 13.52069348054286973758601658316, 16.02633178502293180169785871645, 16.95462444550224700252703829235

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