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Properties

Label 1-1205-1205.1089-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.286 - 0.958i$
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  + i·2-s + (−0.951 + 0.309i)3-s − 4-s + (−0.309 − 0.951i)6-s + (0.156 + 0.987i)7-s − i·8-s + (0.809 − 0.587i)9-s + (0.707 + 0.707i)11-s + (0.951 − 0.309i)12-s + (−0.891 − 0.453i)13-s + (−0.987 + 0.156i)14-s + 16-s + (0.987 − 0.156i)17-s + (0.587 + 0.809i)18-s + (−0.707 − 0.707i)19-s + ⋯
L(s)  = 1  + i·2-s + (−0.951 + 0.309i)3-s − 4-s + (−0.309 − 0.951i)6-s + (0.156 + 0.987i)7-s − i·8-s + (0.809 − 0.587i)9-s + (0.707 + 0.707i)11-s + (0.951 − 0.309i)12-s + (−0.891 − 0.453i)13-s + (−0.987 + 0.156i)14-s + 16-s + (0.987 − 0.156i)17-s + (0.587 + 0.809i)18-s + (−0.707 − 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.286 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.286 - 0.958i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1089, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.286 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01319812877 + 0.009827165202i\)
\(L(\frac12)\) \(\approx\) \(0.01319812877 + 0.009827165202i\)
\(L(1)\) \(\approx\) \(0.4792340027 + 0.3568289734i\)
\(L(1)\) \(\approx\) \(0.4792340027 + 0.3568289734i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (-0.951 + 0.309i)T \)
7 \( 1 + (0.156 + 0.987i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-0.891 - 0.453i)T \)
17 \( 1 + (0.987 - 0.156i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.453 - 0.891i)T \)
29 \( 1 + (-0.587 - 0.809i)T \)
31 \( 1 + (-0.987 + 0.156i)T \)
37 \( 1 + (-0.891 + 0.453i)T \)
41 \( 1 + (-0.587 + 0.809i)T \)
43 \( 1 + (-0.156 + 0.987i)T \)
47 \( 1 + (0.587 + 0.809i)T \)
53 \( 1 + (-0.587 - 0.809i)T \)
59 \( 1 + (-0.951 - 0.309i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (0.587 - 0.809i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (-0.891 + 0.453i)T \)
79 \( 1 + (-0.951 - 0.309i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−21.4765529963465102609710591489, −20.591200873703497315831882799709, −19.6839614503334008666814785552, −19.026101711819690529428714788664, −18.5110900596019983525567685727, −17.28108699606543126953891651633, −17.044188885461156358032659492864, −16.426386171779168380998893286550, −14.884778017669058737584860113277, −14.06427385056144010936553707249, −13.47357466266215811241051746072, −12.39797388892718033397355706033, −12.059518587442868198384692884426, −11.03414894324828200166128373582, −10.616801190087559093311929179025, −9.781367738854945221606849202909, −8.86316153178722906366324511885, −7.669166209673855043762427324791, −7.022384309611779715606905949701, −5.77942957275532706290043113984, −5.101195513664716491163141882815, −4.040461001523561499553911863884, −3.4648068791668133972280732147, −1.85690014430373003258494525573, −1.2516309268406459590933462881, 0.0083056970807686695516968971, 1.5286849035717596190163846391, 3.05644320766328460509484972522, 4.36960493413822122239293401910, 4.93468809992829085173407175733, 5.69079710223286874442108844450, 6.48799373838866738174301795275, 7.21744189141355340250325528127, 8.18229305467072674603354813144, 9.280859457905901548987880898368, 9.698072542793321719173596045223, 10.71433733774400755857021319535, 11.853202652181806695751398237, 12.444544303383344530406853060968, 13.07056685500912268512577997481, 14.615247894848781619380774076076, 14.83610381264460270664753715811, 15.62055437853844000479406560658, 16.46135955243759188248564175229, 17.22841642738196097321983914373, 17.57916362696267252309794432519, 18.5702351627394377829651299663, 19.06130840316794501336900802143, 20.33816549100882067921487038818, 21.48213052823447937247937523828

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