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Properties

Label 2-60e2-100.39-c0-0-0
Degree $2$
Conductor $3600$
Sign $0.992 - 0.125i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
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  + (0.309 + 0.951i)5-s + (1.11 − 1.53i)13-s + (1.11 − 0.363i)17-s + (−0.809 + 0.587i)25-s + (0.5 − 1.53i)29-s + (−0.690 + 0.951i)37-s + (−0.5 − 0.363i)41-s − 49-s + (1.80 + 0.587i)53-s + (0.5 − 0.363i)61-s + (1.80 + 0.587i)65-s + (1.11 + 1.53i)73-s + (0.690 + 0.951i)85-s + (−1.30 + 0.951i)89-s + (−1.11 − 0.363i)97-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)5-s + (1.11 − 1.53i)13-s + (1.11 − 0.363i)17-s + (−0.809 + 0.587i)25-s + (0.5 − 1.53i)29-s + (−0.690 + 0.951i)37-s + (−0.5 − 0.363i)41-s − 49-s + (1.80 + 0.587i)53-s + (0.5 − 0.363i)61-s + (1.80 + 0.587i)65-s + (1.11 + 1.53i)73-s + (0.690 + 0.951i)85-s + (−1.30 + 0.951i)89-s + (−1.11 − 0.363i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.992 - 0.125i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (3439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.992 - 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.473754149\)
\(L(\frac12)\) \(\approx\) \(1.473754149\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.309 - 0.951i)T \)
good7 \( 1 + T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.309 - 0.951i)T^{2} \)
29 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.529616452806252016893212560765, −8.045043293122723865912586328710, −7.28543060669613867635930182436, −6.45492659240324156768767701268, −5.78974993174937988107251461076, −5.19806232696311138521407665250, −3.86325151408085693293601480365, −3.22783399604624739038805683852, −2.44969324779487460457536327300, −1.07918697540379694633948221534, 1.22769238624880365053097598561, 1.93400362703778880498376230326, 3.38560599019448104241631322657, 4.08998780944420589025165940570, 4.97761918195855040222380763838, 5.66394339116937220832616683774, 6.46041069870351156855763937342, 7.17550221094176826541030134784, 8.222891022033781867494455605154, 8.722424573791793826381575772245

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