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Properties

Label 1-2552-2552.531-r1-0-0
Degree $1$
Conductor $2552$
Sign $-0.646 + 0.763i$
Analytic cond. $274.250$
Root an. cond. $274.250$
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  + (0.393 + 0.919i)3-s + (−0.936 − 0.351i)5-s + (−0.753 + 0.657i)7-s + (−0.691 + 0.722i)9-s + (−0.134 − 0.990i)13-s + (−0.0448 − 0.998i)15-s + (−0.309 − 0.951i)17-s + (−0.753 − 0.657i)19-s + (−0.900 − 0.433i)21-s + (−0.623 − 0.781i)23-s + (0.753 + 0.657i)25-s + (−0.936 − 0.351i)27-s + (−0.550 − 0.834i)31-s + (0.936 − 0.351i)35-s + (0.473 + 0.880i)37-s + ⋯
L(s)  = 1  + (0.393 + 0.919i)3-s + (−0.936 − 0.351i)5-s + (−0.753 + 0.657i)7-s + (−0.691 + 0.722i)9-s + (−0.134 − 0.990i)13-s + (−0.0448 − 0.998i)15-s + (−0.309 − 0.951i)17-s + (−0.753 − 0.657i)19-s + (−0.900 − 0.433i)21-s + (−0.623 − 0.781i)23-s + (0.753 + 0.657i)25-s + (−0.936 − 0.351i)27-s + (−0.550 − 0.834i)31-s + (0.936 − 0.351i)35-s + (0.473 + 0.880i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2552 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2552 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.646 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2552\)    =    \(2^{3} \cdot 11 \cdot 29\)
Sign: $-0.646 + 0.763i$
Analytic conductor: \(274.250\)
Root analytic conductor: \(274.250\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2552} (531, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2552,\ (1:\ ),\ -0.646 + 0.763i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1046725077 + 0.2257560674i\)
\(L(\frac12)\) \(\approx\) \(0.1046725077 + 0.2257560674i\)
\(L(1)\) \(\approx\) \(0.7209459164 + 0.1046554333i\)
\(L(1)\) \(\approx\) \(0.7209459164 + 0.1046554333i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.393 + 0.919i)T \)
5 \( 1 + (-0.936 - 0.351i)T \)
7 \( 1 + (-0.753 + 0.657i)T \)
13 \( 1 + (-0.134 - 0.990i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (-0.753 - 0.657i)T \)
23 \( 1 + (-0.623 - 0.781i)T \)
31 \( 1 + (-0.550 - 0.834i)T \)
37 \( 1 + (0.473 + 0.880i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.473 - 0.880i)T \)
53 \( 1 + (0.550 + 0.834i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.995 + 0.0896i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (0.691 + 0.722i)T \)
73 \( 1 + (0.0448 + 0.998i)T \)
79 \( 1 + (-0.691 + 0.722i)T \)
83 \( 1 + (0.858 - 0.512i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
97 \( 1 + (0.995 + 0.0896i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.217679456712041887777568592610, −18.40156253795906653050225448315, −17.70917282918040278592685096384, −16.748710757344970496633946293075, −16.265711119015317890759299070674, −15.30659115916502378085956951400, −14.56232878374885679986295376863, −14.05185998741870270452588130455, −13.13849296845424547716696967601, −12.57614517023449380347474375100, −11.916814525623341336033362276711, −11.10517011898402880128312017975, −10.39773291263072555933291659028, −9.3586950835306238131391649597, −8.6691282925005772403286670523, −7.72012766570542639435796968505, −7.38705006196378245291080299184, −6.40969875139952524954381246769, −6.104550559190720029700648034941, −4.47335945383490553885185192947, −3.80756300104604437674036467101, −3.1847679029439910351804861734, −2.116135430737010613668917251412, −1.26885197391375602533213469631, −0.071181605583127885400146486608, 0.505767598321285710239641477523, 2.32855032147230995751859309378, 2.894393927919247474290829664693, 3.741860480281819003224075734636, 4.47935884435518165764262404937, 5.21599478064331145565566068085, 6.03331361957264804736882511903, 7.11445411688945623727346396254, 7.96457682812308467729295613906, 8.68812143158495420784470126158, 9.19041255852544629222894365461, 10.03282748119819772024846688453, 10.77061984482446732923793702283, 11.53757079408741707907570777293, 12.29896498600028491572074169909, 13.001793491407155026528934950534, 13.78544919348272475341585591975, 14.836763919083106436533030872102, 15.42217493583877100260735402655, 15.67844383758116305154327597380, 16.55495090969432250884787939745, 17.038204717902260454774371826279, 18.27909631462577735157481799912, 18.84327091059661333195765464287, 19.797800500609258792378982265271

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