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Properties

Label 1-675-675.571-r0-0-0
Degree $1$
Conductor $675$
Sign $0.691 - 0.722i$
Analytic cond. $3.13468$
Root an. cond. $3.13468$
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  + (−0.241 − 0.970i)2-s + (−0.882 + 0.469i)4-s + (0.173 − 0.984i)7-s + (0.669 + 0.743i)8-s + (0.961 + 0.275i)11-s + (−0.241 + 0.970i)13-s + (−0.997 + 0.0697i)14-s + (0.559 − 0.829i)16-s + (−0.978 + 0.207i)17-s + (0.669 + 0.743i)19-s + (0.0348 − 0.999i)22-s + (0.438 − 0.898i)23-s + 26-s + (0.309 + 0.951i)28-s + (−0.374 + 0.927i)29-s + ⋯
L(s)  = 1  + (−0.241 − 0.970i)2-s + (−0.882 + 0.469i)4-s + (0.173 − 0.984i)7-s + (0.669 + 0.743i)8-s + (0.961 + 0.275i)11-s + (−0.241 + 0.970i)13-s + (−0.997 + 0.0697i)14-s + (0.559 − 0.829i)16-s + (−0.978 + 0.207i)17-s + (0.669 + 0.743i)19-s + (0.0348 − 0.999i)22-s + (0.438 − 0.898i)23-s + 26-s + (0.309 + 0.951i)28-s + (−0.374 + 0.927i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(3.13468\)
Root analytic conductor: \(3.13468\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 675,\ (0:\ ),\ 0.691 - 0.722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.087221350 - 0.4644869866i\)
\(L(\frac12)\) \(\approx\) \(1.087221350 - 0.4644869866i\)
\(L(1)\) \(\approx\) \(0.8755064143 - 0.3616186500i\)
\(L(1)\) \(\approx\) \(0.8755064143 - 0.3616186500i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.241 - 0.970i)T \)
7 \( 1 + (0.173 - 0.984i)T \)
11 \( 1 + (0.961 + 0.275i)T \)
13 \( 1 + (-0.241 + 0.970i)T \)
17 \( 1 + (-0.978 + 0.207i)T \)
19 \( 1 + (0.669 + 0.743i)T \)
23 \( 1 + (0.438 - 0.898i)T \)
29 \( 1 + (-0.374 + 0.927i)T \)
31 \( 1 + (0.990 + 0.139i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
41 \( 1 + (-0.241 + 0.970i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.990 - 0.139i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.961 - 0.275i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (-0.374 - 0.927i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (-0.374 + 0.927i)T \)
83 \( 1 + (0.848 - 0.529i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (-0.615 - 0.788i)T \)
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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

−22.76825148313842193704213071255, −22.241392089699098622174462397305, −21.56504828093912176280599749534, −20.20902039690687605168916232417, −19.3993417066082384846591727796, −18.64758130316421113095820952440, −17.61818680513423073147801251467, −17.3660893031208283020461815954, −16.11488815514509745820642235016, −15.36486625524701665530498276688, −14.935397888272832335108344991452, −13.79289299352275778793004701174, −13.14108737825581273247345044653, −11.96467274616541513322638887555, −11.1316191271911302214786830804, −9.82104039374947820975804894334, −9.133937675222144480818551075777, −8.41152821076743112051964027318, −7.43728700350658092628154158395, −6.49694886809782685944437053433, −5.628320846810638151268058548254, −4.88533887972230610480720961150, −3.68755965367790681543813289862, −2.3485283894400376294092661697, −0.82787824361203001184333399951, 1.04906910060096762654362486104, 1.92186290401012370900037222898, 3.22681319069799939476029872684, 4.2420011408179600439854305342, 4.72956739516982246789216451000, 6.42726318433195310385842350719, 7.29357581813108911135780321824, 8.38221443384672835107478812295, 9.26090172578043302691949568311, 10.02838049520982268077224922727, 10.92782700440828601346265450845, 11.655210632925307585184797189948, 12.45364792221950871165711882840, 13.49204468428195312428229037455, 14.110502355976758780466645887865, 14.928463731413839984838469193438, 16.63465807283220395053200133877, 16.81005153194826162955740476003, 17.88134778297149856719696114435, 18.62045602088316131190652745650, 19.65377611784084547566542894380, 20.07942944176697070638948051273, 20.866798641882822469167825373168, 21.7892202561642467185249081474, 22.48924031119137724652634432559

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