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Properties

Label 1-1205-1205.1003-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.663 - 0.748i$
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  + (0.965 + 0.258i)2-s + (0.838 − 0.544i)3-s + (0.866 + 0.5i)4-s + (0.951 − 0.309i)6-s + (−0.284 − 0.958i)7-s + (0.707 + 0.707i)8-s + (0.406 − 0.913i)9-s + (0.608 − 0.793i)11-s + (0.998 − 0.0523i)12-s + (0.824 + 0.566i)13-s + (−0.0261 − 0.999i)14-s + (0.5 + 0.866i)16-s + (−0.522 − 0.852i)17-s + (0.629 − 0.777i)18-s + (−0.793 − 0.608i)19-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.838 − 0.544i)3-s + (0.866 + 0.5i)4-s + (0.951 − 0.309i)6-s + (−0.284 − 0.958i)7-s + (0.707 + 0.707i)8-s + (0.406 − 0.913i)9-s + (0.608 − 0.793i)11-s + (0.998 − 0.0523i)12-s + (0.824 + 0.566i)13-s + (−0.0261 − 0.999i)14-s + (0.5 + 0.866i)16-s + (−0.522 − 0.852i)17-s + (0.629 − 0.777i)18-s + (−0.793 − 0.608i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.663 - 0.748i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.663 - 0.748i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.603432335 - 1.621830962i\)
\(L(\frac12)\) \(\approx\) \(3.603432335 - 1.621830962i\)
\(L(1)\) \(\approx\) \(2.429115781 - 0.4677408765i\)
\(L(1)\) \(\approx\) \(2.429115781 - 0.4677408765i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (0.838 - 0.544i)T \)
7 \( 1 + (-0.284 - 0.958i)T \)
11 \( 1 + (0.608 - 0.793i)T \)
13 \( 1 + (0.824 + 0.566i)T \)
17 \( 1 + (-0.522 - 0.852i)T \)
19 \( 1 + (-0.793 - 0.608i)T \)
23 \( 1 + (-0.649 - 0.760i)T \)
29 \( 1 + (-0.933 + 0.358i)T \)
31 \( 1 + (0.878 + 0.477i)T \)
37 \( 1 + (-0.824 + 0.566i)T \)
41 \( 1 + (-0.156 - 0.987i)T \)
43 \( 1 + (0.233 + 0.972i)T \)
47 \( 1 + (-0.987 - 0.156i)T \)
53 \( 1 + (0.777 + 0.629i)T \)
59 \( 1 + (0.838 + 0.544i)T \)
61 \( 1 + (0.891 + 0.453i)T \)
67 \( 1 + (0.629 + 0.777i)T \)
71 \( 1 + (-0.725 + 0.688i)T \)
73 \( 1 + (0.0784 + 0.996i)T \)
79 \( 1 + (0.891 - 0.453i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (0.991 - 0.130i)T \)
97 \( 1 + (-0.104 - 0.994i)T \)
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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

−21.13609076315392643546413202387, −20.79632496808580751235546017412, −19.78745524620198828881612509503, −19.3712319354548392575253372487, −18.514472904405225021838276265138, −17.33858965538597913028423044744, −16.25192406894830067449886085076, −15.507041010139344240626677455166, −15.0663946237850190917546364004, −14.47116976165469647117929825248, −13.41027502680777152515284155430, −12.90427145481299950112900992575, −12.053149420943651656324279969592, −11.159445928737042186165613242170, −10.23527764659739075804720133349, −9.59388422523032998808615868024, −8.602987934140930893491993992397, −7.81642402181620735152206692959, −6.57780832930557735900249520537, −5.86304352268685523787503969948, −4.919984331757688950039866699113, −3.87336142696443614296829321725, −3.50482220388977283391027616785, −2.194028859832666873443004849, −1.802022596330251850996229322617, 1.031206121863510082862400606807, 2.13875279181625192475920255228, 3.12994897812844951381173795960, 3.88580034802450144006690328536, 4.51550004758396473068480256560, 5.96954862176166884496806168201, 6.77859495567824720461591975049, 7.09632619289062693837639341966, 8.34990448050061501740018152187, 8.83342373347558446251002482409, 10.11736788537006303099222722860, 11.1449701906182588225402493070, 11.79438639290914088250713220854, 12.88831737066783435523589457969, 13.43395380160556816468476675765, 14.02337944092274433341592693079, 14.5276132771042964856696206665, 15.61887267223267333143335257917, 16.25269855724945563986316916679, 17.04668012543325997466014487684, 17.98954489724510730787089092239, 19.09495158939680630364572188084, 19.62748141797453363889850898570, 20.4819659063387960385751403279, 20.9086171914345827027305934763

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