Properties

Label 1-2205-2205.1139-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.0889 - 0.996i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)8-s + (−0.365 + 0.930i)11-s + (0.365 − 0.930i)13-s + (−0.900 − 0.433i)16-s + (−0.955 − 0.294i)17-s + (0.5 + 0.866i)19-s + (−0.955 + 0.294i)22-s + (−0.733 − 0.680i)23-s + (0.955 − 0.294i)26-s + (−0.955 − 0.294i)29-s − 31-s + (−0.222 − 0.974i)32-s + (−0.365 − 0.930i)34-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (−0.900 + 0.433i)8-s + (−0.365 + 0.930i)11-s + (0.365 − 0.930i)13-s + (−0.900 − 0.433i)16-s + (−0.955 − 0.294i)17-s + (0.5 + 0.866i)19-s + (−0.955 + 0.294i)22-s + (−0.733 − 0.680i)23-s + (0.955 − 0.294i)26-s + (−0.955 − 0.294i)29-s − 31-s + (−0.222 − 0.974i)32-s + (−0.365 − 0.930i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0889 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0889 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.0889 - 0.996i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.0889 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1342378984 - 0.1227866455i\)
\(L(\frac12)\) \(\approx\) \(0.1342378984 - 0.1227866455i\)
\(L(1)\) \(\approx\) \(0.9149171752 + 0.4850138945i\)
\(L(1)\) \(\approx\) \(0.9149171752 + 0.4850138945i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.365 + 0.930i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (-0.955 - 0.294i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.733 - 0.680i)T \)
29 \( 1 + (-0.955 - 0.294i)T \)
31 \( 1 - T \)
37 \( 1 + (0.733 - 0.680i)T \)
41 \( 1 + (0.826 + 0.563i)T \)
43 \( 1 + (-0.826 + 0.563i)T \)
47 \( 1 + (-0.623 - 0.781i)T \)
53 \( 1 + (-0.733 - 0.680i)T \)
59 \( 1 + (-0.900 - 0.433i)T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 - T \)
71 \( 1 + (0.222 - 0.974i)T \)
73 \( 1 + (0.365 + 0.930i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.365 - 0.930i)T \)
89 \( 1 + (-0.988 - 0.149i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−19.92174358506036285116105083482, −19.36913004799452812316363612020, −18.51060582488623514411108592943, −18.10031491615206840552998445115, −17.02752732816332624095547519777, −16.08123494497429345245827341710, −15.541377794767887194002087780396, −14.67266144605780436162759832717, −13.81268033162807134318081168, −13.45369762752949240965098388500, −12.696091768673366807244850661572, −11.73371237900451124203183893126, −11.12064786357049408104738784530, −10.75027656744568327444969892507, −9.46749392649827550138629175402, −9.158286683577866569429041537670, −8.136306827445918918210399128734, −7.0131136304949644175802625725, −6.18691099772742601096680377614, −5.508403562040920974366193197135, −4.604429768095385190666157055333, −3.83454831200703043674695563710, −3.0759358302242208742195241336, −2.12644792899319310360339105675, −1.31838043046875968061223805564, 0.045334743563750794970517016339, 1.8244904805776564063160735668, 2.748281504632777202329309176849, 3.697136107033862319400867672200, 4.439594501368764186810565026893, 5.27249852814955219807316844389, 5.96808321335057472984890058903, 6.76992311380746235949632310704, 7.686504230126611995821105739737, 8.05863600609565234563094534266, 9.12010820906306611106182715050, 9.85644276525880318260973314785, 10.85713145221949436861325474179, 11.67672580956587182678961058140, 12.60523514874056560032566352604, 12.99343945552478578371168663288, 13.76144670528151653758658371903, 14.75956154460134155489767810864, 15.02686850370933740211719920549, 16.030058388165338091736970427479, 16.386752223176334814370719660433, 17.39082461695303640278745472579, 18.2119526517808154630299851595, 18.272661681915464519948267615693, 19.877368475775753370906444406071

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