Properties

Label 2-60e2-300.203-c0-0-0
Degree $2$
Conductor $3600$
Sign $0.965 - 0.261i$
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  + (−0.156 + 0.987i)5-s + (0.896 − 1.76i)13-s + (0.610 + 0.0966i)17-s + (−0.951 − 0.309i)25-s + (0.734 + 0.533i)29-s + (0.809 + 0.412i)37-s + (1.87 − 0.610i)41-s − i·49-s + (−1.59 + 0.253i)53-s + (−0.363 + 1.11i)61-s + (1.59 + 1.16i)65-s + (0.278 − 0.142i)73-s + (−0.190 + 0.587i)85-s + (−0.550 + 1.69i)89-s + (1.76 − 0.278i)97-s + ⋯
L(s)  = 1  + (−0.156 + 0.987i)5-s + (0.896 − 1.76i)13-s + (0.610 + 0.0966i)17-s + (−0.951 − 0.309i)25-s + (0.734 + 0.533i)29-s + (0.809 + 0.412i)37-s + (1.87 − 0.610i)41-s − i·49-s + (−1.59 + 0.253i)53-s + (−0.363 + 1.11i)61-s + (1.59 + 1.16i)65-s + (0.278 − 0.142i)73-s + (−0.190 + 0.587i)85-s + (−0.550 + 1.69i)89-s + (1.76 − 0.278i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 - 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.965 - 0.261i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.965 - 0.261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.328520601\)
\(L(\frac12)\) \(\approx\) \(1.328520601\)
\(L(1)\) 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 \( 1 + (0.156 - 0.987i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.610 - 0.0966i)T + (0.951 + 0.309i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (-0.734 - 0.533i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (-1.87 + 0.610i)T + (0.809 - 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (1.59 - 0.253i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.951 + 0.309i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (0.550 - 1.69i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
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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

−8.620059380890689167379639811480, −7.87865355469595488223412369863, −7.47120467722528340817611584339, −6.40108964217679267426320692948, −5.92872278955977004202333603488, −5.08809134614581651086690728098, −3.92085286254233371845352983621, −3.22070817995563944591413679833, −2.54111789546927266121644305788, −1.04346522296761098263226343852, 1.07578613619746868521631991607, 2.01004254692297942261471047378, 3.29692793893087266062611180122, 4.35101025796294310236399061414, 4.60213617538896462675438147560, 5.85573819791277670292802779839, 6.29085338100211500341647259895, 7.33840057420360788628017146974, 8.042705549381393225085205833790, 8.725615539769453779287074724004

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