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Properties

Label 2-60e2-36.31-c0-0-1
Degree $2$
Conductor $3600$
Sign $0.173 + 0.984i$
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  + (0.5 − 0.866i)3-s + (1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s − 1.73i·21-s + (1.5 + 0.866i)23-s − 0.999·27-s + (−0.5 − 0.866i)29-s + (−0.5 + 0.866i)41-s + (−1.5 + 0.866i)47-s + (1 − 1.73i)49-s + (−0.5 − 0.866i)61-s + (−1.49 − 0.866i)63-s + (1.5 + 0.866i)67-s + (1.5 − 0.866i)69-s + (−0.5 + 0.866i)81-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (1.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s − 1.73i·21-s + (1.5 + 0.866i)23-s − 0.999·27-s + (−0.5 − 0.866i)29-s + (−0.5 + 0.866i)41-s + (−1.5 + 0.866i)47-s + (1 − 1.73i)49-s + (−0.5 − 0.866i)61-s + (−1.49 − 0.866i)63-s + (1.5 + 0.866i)67-s + (1.5 − 0.866i)69-s + (−0.5 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.786094177\)
\(L(\frac12)\) \(\approx\) \(1.786094177\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
good7 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−8.223318241315047015748702234177, −7.939884712043278586036290269757, −7.19968733023872853066613802129, −6.60735342854946417288903520780, −5.51852554434213728838964364638, −4.77111005575150577678909005913, −3.87043298703629266043180909715, −2.92890576232569559654620344112, −1.79692073878321632397685117391, −1.10428701198058591233431240327, 1.63846770763424277429347827945, 2.51315371053266114050122915054, 3.40271583100355970322259024002, 4.45266255159392665606585548180, 5.08713894358954790474357448754, 5.50827504382956451802312524405, 6.74106956231082083138956396413, 7.64955022906691997211082746268, 8.413552785979424476797980298620, 8.774233293235374318073716510000

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