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Properties

Label 1-2205-2205.2164-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.974 + 0.225i$
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  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)8-s + (−0.733 − 0.680i)11-s + (−0.955 + 0.294i)13-s + (−0.988 + 0.149i)16-s + (0.900 − 0.433i)17-s + 19-s + (−0.0747 − 0.997i)22-s + (−0.0747 − 0.997i)23-s + (−0.900 − 0.433i)26-s + (0.0747 − 0.997i)29-s + (−0.5 − 0.866i)31-s + (−0.826 − 0.563i)32-s + (0.955 + 0.294i)34-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)8-s + (−0.733 − 0.680i)11-s + (−0.955 + 0.294i)13-s + (−0.988 + 0.149i)16-s + (0.900 − 0.433i)17-s + 19-s + (−0.0747 − 0.997i)22-s + (−0.0747 − 0.997i)23-s + (−0.900 − 0.433i)26-s + (0.0747 − 0.997i)29-s + (−0.5 − 0.866i)31-s + (−0.826 − 0.563i)32-s + (0.955 + 0.294i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.974 + 0.225i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2164, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.974 + 0.225i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.016900209 + 0.2308884058i\)
\(L(\frac12)\) \(\approx\) \(2.016900209 + 0.2308884058i\)
\(L(1)\) \(\approx\) \(1.352920903 + 0.4321198108i\)
\(L(1)\) \(\approx\) \(1.352920903 + 0.4321198108i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.733 + 0.680i)T \)
11 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (-0.955 + 0.294i)T \)
17 \( 1 + (0.900 - 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (0.0747 - 0.997i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.900 - 0.433i)T \)
41 \( 1 + (-0.988 - 0.149i)T \)
43 \( 1 + (0.988 - 0.149i)T \)
47 \( 1 + (0.733 + 0.680i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
59 \( 1 + (0.365 - 0.930i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (0.222 + 0.974i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.955 - 0.294i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
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

−19.87157459740197808910834786557, −19.21856724303582995014335604663, −18.25868394254464323590693089836, −17.85763696417309641887471176311, −16.73590220531049128068918792521, −15.93165545493753064511282201226, −15.109686092342700422752055830523, −14.65399252623722893388563847374, −13.796999713020177324910178065274, −13.09530644071958878290651995570, −12.32211586669664796987163532412, −11.92669992241371018043522842408, −10.93648351394155391035084629744, −10.10246433565353980553745546686, −9.79533138591538375480346427214, −8.750069155264787510946456612511, −7.511943513661236630742670614911, −7.11265871736892756650960009001, −5.7674680469726985576875374813, −5.29837161323570676279681359462, −4.57785945136362593384940784565, −3.50122666939585871193412472282, −2.88644220996190248865753209926, −1.9414422252906502505740109197, −1.031731431078688395204806397504, 0.5805596903676344328297361217, 2.31505325371125064980000704455, 2.88202282198939160646391435295, 3.8639001185967131202573273813, 4.70744374283263826344339360720, 5.50181732212272989197233385690, 6.02685907935106811433334109699, 7.162274376198467074429887060415, 7.64043400203685187196652889357, 8.37792607060836919809890153588, 9.357004143880701668317732928, 10.1231211649013362406647649175, 11.24123151204059151640103326745, 11.84602224423212366215975487591, 12.61671965469069791583320623452, 13.28550898337611404751456986379, 14.18643940420445691658176058177, 14.459150775118924457729009179102, 15.52842495525462652181032465203, 16.02842115776978845233945130293, 16.80425179428443627010094942461, 17.28097340745206880202435953791, 18.42677912819573349907505040990, 18.75049885917584142246129062555, 19.963973207448532992054808544829

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