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Properties

Label 1-1205-1205.1054-r0-0-0
Degree $1$
Conductor $1205$
Sign $-0.964 + 0.262i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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  + (−0.5 + 0.866i)2-s + (−0.104 + 0.994i)3-s + (−0.5 − 0.866i)4-s + (−0.809 − 0.587i)6-s + (−0.743 + 0.669i)7-s + 8-s + (−0.978 − 0.207i)9-s + (0.866 + 0.5i)11-s + (0.913 − 0.406i)12-s + (0.994 + 0.104i)13-s + (−0.207 − 0.978i)14-s + (−0.5 + 0.866i)16-s + (−0.951 − 0.309i)17-s + (0.669 − 0.743i)18-s + (0.866 + 0.5i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.104 + 0.994i)3-s + (−0.5 − 0.866i)4-s + (−0.809 − 0.587i)6-s + (−0.743 + 0.669i)7-s + 8-s + (−0.978 − 0.207i)9-s + (0.866 + 0.5i)11-s + (0.913 − 0.406i)12-s + (0.994 + 0.104i)13-s + (−0.207 − 0.978i)14-s + (−0.5 + 0.866i)16-s + (−0.951 − 0.309i)17-s + (0.669 − 0.743i)18-s + (0.866 + 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $-0.964 + 0.262i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1054, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ -0.964 + 0.262i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1320878740 + 0.9871316131i\)
\(L(\frac12)\) \(\approx\) \(0.1320878740 + 0.9871316131i\)
\(L(1)\) \(\approx\) \(0.5337608932 + 0.5789887642i\)
\(L(1)\) \(\approx\) \(0.5337608932 + 0.5789887642i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.104 + 0.994i)T \)
7 \( 1 + (-0.743 + 0.669i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.994 + 0.104i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + (0.587 - 0.809i)T \)
29 \( 1 + (0.978 + 0.207i)T \)
31 \( 1 + (-0.743 + 0.669i)T \)
37 \( 1 + (0.994 - 0.104i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (0.669 - 0.743i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (0.207 - 0.978i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (-0.866 - 0.5i)T \)
97 \( 1 + (-0.669 + 0.743i)T \)
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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

−20.34838994533478924996158538246, −20.019733821711233429865497147676, −19.277449062912970436988876344763, −18.712798306752200533366762099591, −17.81968228021323550547460038348, −17.25558639878400171169983114246, −16.53406605493846964080811888201, −15.64226592051546392026470442430, −14.1122378521275143680735002534, −13.53186511734173061112446415354, −13.07138452042370900057075954703, −12.18937777468534344959088535945, −11.21607286991580916382864268064, −10.992866708339022055881154455080, −9.68431397703893892489384303333, −8.97145839637146306663553598498, −8.2242764692019505707848038733, −7.210420353576864892751507752298, −6.62312440091892072605245326467, −5.58969624330598967116891212825, −4.11604848389927097895947579737, −3.410373117423818362130885149297, −2.48093571394245826771994945548, −1.3030800304877012076808071290, −0.64254218553597910871994930430, 1.0705768494313479598451285167, 2.59762584193997210626299342408, 3.77676837958285316167077695330, 4.58218416515116774805568626806, 5.520206249010579546191154519152, 6.30319797179554050597448265858, 6.912911202539206661290618491247, 8.2788123869467897581552505048, 9.043042213209186977550187482960, 9.38562595428915381325252866336, 10.30201504380151236212534859821, 11.10361654892999957207805867593, 12.032634413674108260686534089786, 13.16207747214753581430948496598, 14.12097362720681542597169409059, 14.817527002401675400951420611590, 15.510257391715580391817386370147, 16.297062347680536027480483796047, 16.481367635081860475183666497961, 17.74147185323222241253669517343, 18.14413547367936841396472265536, 19.21877379927072605231544127471, 19.92227677677802810857285906176, 20.63807097232778647745493529392, 21.75688306561671893895075147168

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