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Properties

Label 1-60e2-3600.1237-r1-0-0
Degree $1$
Conductor $3600$
Sign $0.826 + 0.563i$
Analytic cond. $386.873$
Root an. cond. $386.873$
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  + (0.866 + 0.5i)7-s + (0.207 − 0.978i)11-s + (0.978 − 0.207i)13-s + (0.587 − 0.809i)17-s + (0.587 − 0.809i)19-s + (0.743 + 0.669i)23-s + (0.994 − 0.104i)29-s + (−0.104 + 0.994i)31-s + (−0.309 + 0.951i)37-s + (0.978 − 0.207i)41-s + (−0.5 + 0.866i)43-s + (0.994 − 0.104i)47-s + (0.5 + 0.866i)49-s + (−0.809 + 0.587i)53-s + (0.207 + 0.978i)59-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)7-s + (0.207 − 0.978i)11-s + (0.978 − 0.207i)13-s + (0.587 − 0.809i)17-s + (0.587 − 0.809i)19-s + (0.743 + 0.669i)23-s + (0.994 − 0.104i)29-s + (−0.104 + 0.994i)31-s + (−0.309 + 0.951i)37-s + (0.978 − 0.207i)41-s + (−0.5 + 0.866i)43-s + (0.994 − 0.104i)47-s + (0.5 + 0.866i)49-s + (−0.809 + 0.587i)53-s + (0.207 + 0.978i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.826 + 0.563i$
Analytic conductor: \(386.873\)
Root analytic conductor: \(386.873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3600,\ (1:\ ),\ 0.826 + 0.563i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.322127483 + 1.025194196i\)
\(L(\frac12)\) \(\approx\) \(3.322127483 + 1.025194196i\)
\(L(1)\) \(\approx\) \(1.420825523 + 0.07244514470i\)
\(L(1)\) \(\approx\) \(1.420825523 + 0.07244514470i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.207 - 0.978i)T \)
13 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (0.587 - 0.809i)T \)
19 \( 1 + (0.587 - 0.809i)T \)
23 \( 1 + (0.743 + 0.669i)T \)
29 \( 1 + (0.994 - 0.104i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.994 - 0.104i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (0.207 + 0.978i)T \)
61 \( 1 + (-0.207 + 0.978i)T \)
67 \( 1 + (-0.104 + 0.994i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.951 + 0.309i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.994 - 0.104i)T \)
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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

−18.51641850372402886048791414701, −17.62764673866111799453039799522, −17.234100334379057304264273216263, −16.4472665413160626035996887690, −15.71989812126470655534987645952, −14.88130952806652266461444766957, −14.35825268056988797413865999847, −13.78582382449908698587168194059, −12.81427626986229902456413373620, −12.28530037257549481115567479024, −11.447328221237829807481509899529, −10.764521633226700052834569397066, −10.18383988542105950438780558408, −9.348588461800779295776798528111, −8.49357166269456203032211887353, −7.87023582620703542610580433217, −7.19912402430051889502911618261, −6.35026684831131423791136904181, −5.57022506075922039460486371814, −4.68185213743457470755611796449, −4.052244105749344882855918620737, −3.32186095498427114363230954026, −2.0623884243066068176505847699, −1.4735750015562914693708428823, −0.60230025989654362538932210975, 1.03964807054912572948342767912, 1.19075248862261459851874519465, 2.71906538096502769950922088496, 3.111620754130495747906276732053, 4.18311098503904765166780376147, 5.1097897686522116005125855562, 5.57724618033456868616412124923, 6.444462157636806659068460139082, 7.31571575951189388155229856939, 8.08954347765321640146840138064, 8.78024853496114080514377877014, 9.24329556146271962063829902955, 10.330572991626499704144118330283, 11.08681663833879147355382549771, 11.55737975729201901680015303111, 12.17961581573377383857808219906, 13.215991582784209414331640677597, 13.84904983166544399394438602335, 14.29614632827455747645801910172, 15.27944089864142743778107616586, 15.81358549074241636751854326062, 16.42849210732024460077340282814, 17.34624932796066419444815272988, 17.957431258108431907136190774516, 18.50581603025759531563496805232

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