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Properties

Label 2-28-28.27-c1-0-1
Degree $2$
Conductor $28$
Sign $0.661 + 0.750i$
Analytic cond. $0.223581$
Root an. cond. $0.472843$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 1.32i)2-s + (−1.50 + 1.32i)4-s + 2.64i·7-s + (2.50 + 1.32i)8-s − 3·9-s − 5.29i·11-s + (3.50 − 1.32i)14-s + (0.500 − 3.96i)16-s + (1.5 + 3.96i)18-s + (−7.00 + 2.64i)22-s + 5.29i·23-s + 5·25-s + (−3.50 − 3.96i)28-s − 2·29-s + (−5.50 + 1.32i)32-s + ⋯
L(s)  = 1  + (−0.353 − 0.935i)2-s + (−0.750 + 0.661i)4-s + 0.999i·7-s + (0.883 + 0.467i)8-s − 9-s − 1.59i·11-s + (0.935 − 0.353i)14-s + (0.125 − 0.992i)16-s + (0.353 + 0.935i)18-s + (−1.49 + 0.564i)22-s + 1.10i·23-s + 25-s + (−0.661 − 0.749i)28-s − 0.371·29-s + (−0.972 + 0.233i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28\)    =    \(2^{2} \cdot 7\)
Sign: $0.661 + 0.750i$
Analytic conductor: \(0.223581\)
Root analytic conductor: \(0.472843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{28} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 28,\ (\ :1/2),\ 0.661 + 0.750i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506673 - 0.228720i\)
\(L(\frac12)\) \(\approx\) \(0.506673 - 0.228720i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 1.32i)T \)
7 \( 1 - 2.64iT \)
good3 \( 1 + 3T^{2} \)
5 \( 1 - 5T^{2} \)
11 \( 1 + 5.29iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 5.29iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 5.29iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 15.8iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−17.36103568531052980967719794657, −16.20713218939645642377157653893, −14.46184169942106912708858944273, −13.24557983896369712639090803793, −11.80582065632319781891202715000, −11.00726626863206248848760142427, −9.195135941073282046423675747516, −8.288881670273938005137198112079, −5.61837662151843458306893694869, −3.05576516303848460288434731668, 4.67922456886986568351537135220, 6.62673790339788372172641120178, 7.916820437477627041447330894700, 9.487120310487436012715062545611, 10.77438049141535870105484223016, 12.76869743899601511049677332802, 14.18991637382708731070571093951, 14.98415704814163214532725277147, 16.51354540129250993514876248337, 17.28002321670383671839280006782

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