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Properties

Label 1-459-459.275-r0-0-0
Degree $1$
Conductor $459$
Sign $-0.284 + 0.958i$
Analytic cond. $2.13158$
Root an. cond. $2.13158$
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  + (0.573 + 0.819i)2-s + (−0.342 + 0.939i)4-s + (−0.300 − 0.953i)5-s + (0.675 + 0.737i)7-s + (−0.965 + 0.258i)8-s + (0.608 − 0.793i)10-s + (0.953 + 0.300i)11-s + (−0.984 + 0.173i)13-s + (−0.216 + 0.976i)14-s + (−0.766 − 0.642i)16-s + (0.258 + 0.965i)19-s + (0.999 + 0.0436i)20-s + (0.300 + 0.953i)22-s + (0.999 − 0.0436i)23-s + (−0.819 + 0.573i)25-s + (−0.707 − 0.707i)26-s + ⋯
L(s)  = 1  + (0.573 + 0.819i)2-s + (−0.342 + 0.939i)4-s + (−0.300 − 0.953i)5-s + (0.675 + 0.737i)7-s + (−0.965 + 0.258i)8-s + (0.608 − 0.793i)10-s + (0.953 + 0.300i)11-s + (−0.984 + 0.173i)13-s + (−0.216 + 0.976i)14-s + (−0.766 − 0.642i)16-s + (0.258 + 0.965i)19-s + (0.999 + 0.0436i)20-s + (0.300 + 0.953i)22-s + (0.999 − 0.0436i)23-s + (−0.819 + 0.573i)25-s + (−0.707 − 0.707i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.284 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(459\)    =    \(3^{3} \cdot 17\)
Sign: $-0.284 + 0.958i$
Analytic conductor: \(2.13158\)
Root analytic conductor: \(2.13158\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{459} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 459,\ (0:\ ),\ -0.284 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.004160325 + 1.344830279i\)
\(L(\frac12)\) \(\approx\) \(1.004160325 + 1.344830279i\)
\(L(1)\) \(\approx\) \(1.156329546 + 0.7151437368i\)
\(L(1)\) \(\approx\) \(1.156329546 + 0.7151437368i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 + (0.573 + 0.819i)T \)
5 \( 1 + (-0.300 - 0.953i)T \)
7 \( 1 + (0.675 + 0.737i)T \)
11 \( 1 + (0.953 + 0.300i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (0.258 + 0.965i)T \)
23 \( 1 + (0.999 - 0.0436i)T \)
29 \( 1 + (0.843 + 0.537i)T \)
31 \( 1 + (0.737 + 0.675i)T \)
37 \( 1 + (-0.130 - 0.991i)T \)
41 \( 1 + (-0.843 + 0.537i)T \)
43 \( 1 + (0.0871 + 0.996i)T \)
47 \( 1 + (0.342 + 0.939i)T \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (0.996 + 0.0871i)T \)
61 \( 1 + (-0.675 - 0.737i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (0.130 + 0.991i)T \)
73 \( 1 + (-0.608 - 0.793i)T \)
79 \( 1 + (0.216 + 0.976i)T \)
83 \( 1 + (-0.573 - 0.819i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (0.887 - 0.461i)T \)
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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

−23.58038744440644170783039385097, −22.550095270916100429698929500749, −22.132257274137941497181445894319, −21.19565338656102996525338433449, −20.22315243793395817816487548686, −19.47533989560241244452156626295, −18.886212380943245324960275159920, −17.707093376910709509065644170163, −17.0685056232115682606995876002, −15.42265354315623982510857835087, −14.819446446083299369193308105845, −14.01320815896177135115932851406, −13.369971772134005566240940606509, −11.8923534845311525996526505012, −11.52399113369034804100424387370, −10.546140757095119712151563147909, −9.85102248515803096476342384372, −8.60285227610208211392850022250, −7.23952674469390768674264305104, −6.52629554490774770832249300454, −5.11614311248013930210547686216, −4.239531112667233607340168098229, −3.260779178794813611407305346754, −2.27855794874800065614564182815, −0.85411027430445749355307010885, 1.51066520794002376828882651322, 3.03614654285353817936695933039, 4.409057584670732294256323709598, 4.91754210362623910353254977310, 5.91347563861849227052090382160, 7.07555072461720559133687329815, 8.0425930493791099522641686856, 8.814652454960131399347556738479, 9.59906130426927174841991047655, 11.40322082921900541763015051617, 12.27573211873950802448796562877, 12.57061995097608168092255778177, 14.0158891192222513793815727538, 14.63312151221991549174907842787, 15.476950304358501146354657494753, 16.381643708918724682999003590895, 17.15237148253382682910801064104, 17.79640553620923934833318024670, 19.040671401995438643991273938413, 20.05534916098829296905218171811, 21.05685415991254048487663687673, 21.63602854596751449493532320411, 22.624120582925699441800704272764, 23.42950476741092514188380498766, 24.38262275701212971424676698747

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