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Properties

Label 1-552-552.125-r0-0-0
Degree $1$
Conductor $552$
Sign $-0.771 + 0.635i$
Analytic cond. $2.56347$
Root an. cond. $2.56347$
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  + (0.654 + 0.755i)5-s + (−0.841 − 0.540i)7-s + (0.142 + 0.989i)11-s + (−0.841 + 0.540i)13-s + (−0.959 + 0.281i)17-s + (−0.959 − 0.281i)19-s + (−0.142 + 0.989i)25-s + (−0.959 + 0.281i)29-s + (0.415 − 0.909i)31-s + (−0.142 − 0.989i)35-s + (−0.654 + 0.755i)37-s + (0.654 + 0.755i)41-s + (0.415 + 0.909i)43-s − 47-s + (0.415 + 0.909i)49-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)5-s + (−0.841 − 0.540i)7-s + (0.142 + 0.989i)11-s + (−0.841 + 0.540i)13-s + (−0.959 + 0.281i)17-s + (−0.959 − 0.281i)19-s + (−0.142 + 0.989i)25-s + (−0.959 + 0.281i)29-s + (0.415 − 0.909i)31-s + (−0.142 − 0.989i)35-s + (−0.654 + 0.755i)37-s + (0.654 + 0.755i)41-s + (0.415 + 0.909i)43-s − 47-s + (0.415 + 0.909i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $-0.771 + 0.635i$
Analytic conductor: \(2.56347\)
Root analytic conductor: \(2.56347\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 552,\ (0:\ ),\ -0.771 + 0.635i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2302173368 + 0.6416185622i\)
\(L(\frac12)\) \(\approx\) \(0.2302173368 + 0.6416185622i\)
\(L(1)\) \(\approx\) \(0.8069817514 + 0.2435828713i\)
\(L(1)\) \(\approx\) \(0.8069817514 + 0.2435828713i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
23 \( 1 \)
good5 \( 1 + (0.654 + 0.755i)T \)
7 \( 1 + (-0.841 - 0.540i)T \)
11 \( 1 + (0.142 + 0.989i)T \)
13 \( 1 + (-0.841 + 0.540i)T \)
17 \( 1 + (-0.959 + 0.281i)T \)
19 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (-0.959 + 0.281i)T \)
31 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (-0.654 + 0.755i)T \)
41 \( 1 + (0.654 + 0.755i)T \)
43 \( 1 + (0.415 + 0.909i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (0.415 - 0.909i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (0.142 - 0.989i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (-0.841 + 0.540i)T \)
83 \( 1 + (0.654 - 0.755i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.654 + 0.755i)T \)
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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

−22.86261875221676408639196639586, −22.09974091493827133028197578458, −21.46435406086866535765895189443, −20.57783502891801318094009282200, −19.5740442147826195750098002096, −19.03347642377430445372180306022, −17.84648488008693784167535661574, −17.12964414207497260289120259407, −16.27993932924254615382801725529, −15.581593501227520986414570861171, −14.46773839366717187328210556790, −13.48137170622618276551590499368, −12.79650454927591794325289381379, −12.097163379697178786773365589188, −10.85631942081776315473453426495, −9.91815157268974333104490382680, −9.0146688477846093498879789725, −8.47700620240046439531320840562, −7.0513820943677336866825497200, −6.01379672615853971672800154227, −5.39504061885754040352957462317, −4.19097288280508542456569711531, −2.91664781413883832691787363710, −1.9633533444643574876626447548, −0.32175635978478863966111222916, 1.83514330821452539577741722318, 2.65862197424166185620737549667, 3.920121073005825203227854621432, 4.866227874696504980694192391451, 6.380913186859895896737465588582, 6.72072020192025389286042038036, 7.714295242815400897509013489504, 9.24810398237793033972035088827, 9.782851469134712360997024286604, 10.62038235850852857118019901571, 11.558788500557622699102885383227, 12.85784945807876751673923451584, 13.28032338453383291320871506112, 14.5071843349887255950303923610, 14.98559687466388989118226135791, 16.11561064309151639039071539120, 17.24447004795171566825013242000, 17.52829985298061742062479160520, 18.7847550168487870689468344625, 19.4318095819081647522946530027, 20.27447166860513593919865189119, 21.27419126014501385331820201804, 22.246414433749714110169090372683, 22.58836071182761227201963427638, 23.5932762900716293404320776417

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