Properties

Label 1-1148-1148.55-r1-0-0
Degree $1$
Conductor $1148$
Sign $0.652 + 0.758i$
Analytic cond. $123.369$
Root an. cond. $123.369$
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  + (−0.707 + 0.707i)3-s + i·5-s i·9-s + (−0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.707 − 0.707i)19-s + 23-s − 25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s − 31-s i·33-s + 37-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + i·5-s i·9-s + (−0.707 + 0.707i)11-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.707 − 0.707i)19-s + 23-s − 25-s + (0.707 + 0.707i)27-s + (−0.707 + 0.707i)29-s − 31-s i·33-s + 37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.652 + 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $0.652 + 0.758i$
Analytic conductor: \(123.369\)
Root analytic conductor: \(123.369\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (1:\ ),\ 0.652 + 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.262269024 + 0.5793011801i\)
\(L(\frac12)\) \(\approx\) \(1.262269024 + 0.5793011801i\)
\(L(1)\) \(\approx\) \(0.7989785081 + 0.2912247120i\)
\(L(1)\) \(\approx\) \(0.7989785081 + 0.2912247120i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + iT \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + T \)
29 \( 1 + (-0.707 + 0.707i)T \)
31 \( 1 - T \)
37 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (0.707 - 0.707i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−21.1255889586890156816031638265, −20.29761661927407915428644863238, −19.18542236390310385792731220698, −18.73242451206478212486985145093, −17.98945592838533890861074675620, −16.941942617637428366252711352165, −16.4799980969429264472689199128, −15.98797884477031861054214846143, −14.68983057443637136758323604612, −13.6509822103245226525315539867, −13.09474120472584620089084271085, −12.48986854816622871758678909789, −11.48823881899973585749855449634, −11.05778706244029376322445599077, −9.89308136477697959813898717787, −8.91828701222316917527424264949, −8.10419793869488328975615501318, −7.41991565727493678115759375390, −6.24671810263142705699255549679, −5.62461902928544482117148228851, −4.85850341286332895939978560121, −3.8266763141441341809053420341, −2.46315377333633836518764276248, −1.37623457929680298683560906450, −0.63410421326806491432139712210, 0.51857977582202249161269896264, 2.01662420823217059566677239611, 3.210132937928844859210131663344, 3.83841197561569167454887239303, 5.03750837114087759914104590095, 5.70330493516687866805905249043, 6.63822248536724095398462841966, 7.37542179979173999543261892995, 8.47295427841554583925359406544, 9.53262409423403823807528537421, 10.37293841761625737171287910652, 10.81326799634502015120458918771, 11.47528744465798961112352576070, 12.65633729745612305925351981545, 13.18575204234222786416328870062, 14.62355953300472143408537079204, 15.02045909509059865646659857204, 15.64167208675342953144627755919, 16.60590311005935088513645354450, 17.38348988777470774129870744772, 18.131689847317947979917985365636, 18.62521377957535958175681967178, 19.728554102925741567962698221621, 20.63471599712798593984677080844, 21.32271402539628012572975812458

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