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Properties

Label 1-15e2-225.121-r0-0-0
Degree $1$
Conductor $225$
Sign $0.747 - 0.663i$
Analytic cond. $1.04489$
Root an. cond. $1.04489$
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  + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (−0.5 + 0.866i)7-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)11-s + (0.913 + 0.406i)13-s + (−0.104 + 0.994i)14-s + (−0.104 − 0.994i)16-s + (0.309 − 0.951i)17-s + (0.309 − 0.951i)19-s + (0.669 − 0.743i)22-s + (−0.104 + 0.994i)23-s + 26-s + (0.309 + 0.951i)28-s + (−0.978 − 0.207i)29-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (0.669 − 0.743i)4-s + (−0.5 + 0.866i)7-s + (0.309 − 0.951i)8-s + (0.913 − 0.406i)11-s + (0.913 + 0.406i)13-s + (−0.104 + 0.994i)14-s + (−0.104 − 0.994i)16-s + (0.309 − 0.951i)17-s + (0.309 − 0.951i)19-s + (0.669 − 0.743i)22-s + (−0.104 + 0.994i)23-s + 26-s + (0.309 + 0.951i)28-s + (−0.978 − 0.207i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.747 - 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $0.747 - 0.663i$
Analytic conductor: \(1.04489\)
Root analytic conductor: \(1.04489\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{225} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 225,\ (0:\ ),\ 0.747 - 0.663i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.923891617 - 0.7308178684i\)
\(L(\frac12)\) \(\approx\) \(1.923891617 - 0.7308178684i\)
\(L(1)\) \(\approx\) \(1.692086655 - 0.4425083568i\)
\(L(1)\) \(\approx\) \(1.692086655 - 0.4425083568i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.913 + 0.406i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-0.978 + 0.207i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.978 - 0.207i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.978 - 0.207i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−26.174452936578098447244375765007, −25.59741556939452554104274810534, −24.589800354276558734029230002094, −23.64826590235271729510277773254, −22.77364658421999629808797469353, −22.27386006942548825551015461995, −20.88636297012240642360167538502, −20.30908845328711424689999709272, −19.24987871515148324451058695850, −17.79068624334219317638855325236, −16.744804097865886065245728761891, −16.19442269263923272448667080442, −14.876841956813968052622278483064, −14.20744702807549613489898497286, −13.09368524846247898824170340380, −12.41521030081315732427110628340, −11.15036162396811689032944473932, −10.17103496088457605862922574450, −8.64161039074933017488029881531, −7.47961237880103630978376455866, −6.522686095528791287782187065830, −5.59457677074399055319281086693, −4.0109406461929966129584050129, −3.55264696322296973144783207556, −1.69867439906740980975715403322, 1.45735015779119323530797772495, 2.89740849139701802459868692236, 3.81658507897849875300923138298, 5.23419467107112037251158544732, 6.13267631886936271870615344149, 7.13602901605971951321140891565, 8.94138588898265858500773224225, 9.7020394478047239191341972021, 11.32356111876171551341326600251, 11.67886141193994778485692169576, 12.95288860570858869570777906139, 13.74123606335722126673910730410, 14.757726990856252982958579959167, 15.77276350895047685786175830325, 16.466185853903521042292223043807, 18.11055864640834906861318310822, 19.050943075108929371346325611362, 19.82524764209824566177337819017, 20.9167073849065934102352801593, 21.80684588237199921596790746948, 22.44602425596920710780613906171, 23.402391428578316388188568586943, 24.425284192111203953094525327847, 25.167691933801683765100729787256, 26.088302341722632146804133753984

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