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Properties

Label 2-1028-1028.531-c0-0-0
Degree $2$
Conductor $1028$
Sign $-0.276 + 0.960i$
Analytic cond. $0.513038$
Root an. cond. $0.716267$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s + (−1.47 + 1.21i)5-s − i·8-s + (−0.980 + 0.195i)9-s + (−1.21 − 1.47i)10-s + (0.216 − 0.324i)13-s + 16-s + (−0.785 − 0.785i)17-s + (−0.195 − 0.980i)18-s + (1.47 − 1.21i)20-s + (0.519 − 2.61i)25-s + (0.324 + 0.216i)26-s + (−1.38 − 0.275i)29-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (−1.47 + 1.21i)5-s − i·8-s + (−0.980 + 0.195i)9-s + (−1.21 − 1.47i)10-s + (0.216 − 0.324i)13-s + 16-s + (−0.785 − 0.785i)17-s + (−0.195 − 0.980i)18-s + (1.47 − 1.21i)20-s + (0.519 − 2.61i)25-s + (0.324 + 0.216i)26-s + (−1.38 − 0.275i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.276 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1028\)    =    \(2^{2} \cdot 257\)
Sign: $-0.276 + 0.960i$
Analytic conductor: \(0.513038\)
Root analytic conductor: \(0.716267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1028} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1028,\ (\ :0),\ -0.276 + 0.960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1364115686\)
\(L(\frac12)\) \(\approx\) \(0.1364115686\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
257 \( 1 + T \)
good3 \( 1 + (0.980 - 0.195i)T^{2} \)
5 \( 1 + (1.47 - 1.21i)T + (0.195 - 0.980i)T^{2} \)
7 \( 1 + (-0.555 + 0.831i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.216 + 0.324i)T + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + (0.785 + 0.785i)T + iT^{2} \)
19 \( 1 + (0.831 + 0.555i)T^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (1.38 + 0.275i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.923 - 0.382i)T^{2} \)
37 \( 1 + (0.902 - 1.68i)T + (-0.555 - 0.831i)T^{2} \)
41 \( 1 + (1.21 - 0.368i)T + (0.831 - 0.555i)T^{2} \)
43 \( 1 + (-0.980 - 0.195i)T^{2} \)
47 \( 1 + (0.831 + 0.555i)T^{2} \)
53 \( 1 + (0.368 - 0.448i)T + (-0.195 - 0.980i)T^{2} \)
59 \( 1 + (0.382 - 0.923i)T^{2} \)
61 \( 1 + (0.425 - 0.636i)T + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (0.831 - 0.555i)T^{2} \)
73 \( 1 + (0.750 - 1.81i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.382 - 0.923i)T^{2} \)
83 \( 1 + (-0.980 - 0.195i)T^{2} \)
89 \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \)
97 \( 1 + (1.26 + 1.53i)T + (-0.195 + 0.980i)T^{2} \)
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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

−10.78040550743596236661131356071, −9.838455231151804028266615221970, −8.615647458689910495259328003366, −8.197870247039297302495901674781, −7.25665070378478649461983028772, −6.78690081639532747831817001149, −5.76252372684723506360174553325, −4.69030561041476943820429916280, −3.67520415494831569032768690134, −2.88216865134159214637786775699, 0.12597442389847513736200472585, 1.77820330494842889831256155229, 3.39786496623886055132434336596, 4.00866345254556146206895227377, 4.90600080518730310239347852565, 5.79170501349710495199094400945, 7.35825171447069762425282682193, 8.277581933334528861773358820994, 8.820425454047515558542062401989, 9.290714417540304526012914842751

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