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Properties

Label 1-273-273.254-r0-0-0
Degree $1$
Conductor $273$
Sign $-0.575 + 0.818i$
Analytic cond. $1.26780$
Root an. cond. $1.26780$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + i·8-s − 10-s + i·11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + i·19-s + (−0.866 − 0.5i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.866 − 0.5i)31-s + (−0.866 + 0.5i)32-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + i·8-s − 10-s + i·11-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + i·19-s + (−0.866 − 0.5i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (−0.866 − 0.5i)31-s + (−0.866 + 0.5i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $-0.575 + 0.818i$
Analytic conductor: \(1.26780\)
Root analytic conductor: \(1.26780\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 273,\ (0:\ ),\ -0.575 + 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7184183837 + 1.383273754i\)
\(L(\frac12)\) \(\approx\) \(0.7184183837 + 1.383273754i\)
\(L(1)\) \(\approx\) \(1.150521218 + 0.7911837491i\)
\(L(1)\) \(\approx\) \(1.150521218 + 0.7911837491i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + iT \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + iT \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + T \)
67 \( 1 + iT \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−25.03974503025716082840895507636, −24.027380762281739927290664010901, −23.80300020621857229764199047788, −22.602158494235165688357590648963, −21.78417447987983213766850344036, −20.92726948233337114131317192705, −19.84108999972790962539537436699, −19.43878261557436250584866310316, −18.3393832352724309998169211833, −16.82354836608748878160374945076, −15.90015204388587403601423892608, −15.17562139026777956068574196069, −14.075746031319204674880483827566, −13.10041764421225375446026179433, −12.30842685396939975948894247673, −11.30730725085313037741559309042, −10.65998678423349908366451726894, −9.171198137527899472888455826414, −8.13923271743438750661449308349, −6.778425507727217147026340803966, −5.69158656561174460708234202924, −4.511151818100179182253663211649, −3.71655346593903378514733593485, −2.461486744237972519689909984951, −0.80507741753867615900110373448, 2.19151670976953279908633200318, 3.469553738521165438908562281055, 4.343999381536624906130433208552, 5.46848648219539007595755956122, 6.82962172565097513613771044285, 7.430009018850828807143949118091, 8.46663248002737017240893544666, 9.99575562119261546906848631673, 11.30907952768665661884441144916, 11.99891426287116917931672799731, 12.936706861633311750773424864062, 14.11612672087276901917959305608, 14.89737220164990675599419086851, 15.680545484480926135823078837824, 16.47528704929256236850135339776, 17.68476264189066767826740673371, 18.596892095841207445240999123969, 20.00617538616457827211000658, 20.47086962458477741448577709948, 21.83903107572954295185203791346, 22.526498936910717575201054713010, 23.33615929550334492372705676308, 23.934197413124261714440678524937, 25.178595091275356663898536815974, 25.73887803708633100186708263908

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