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Properties

Label 2-60e2-15.14-c0-0-3
Degree $2$
Conductor $3600$
Sign $0.151 + 0.988i$
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  − i·7-s − 1.41i·11-s + i·13-s − 1.41·17-s + 19-s + 1.41·23-s − 1.41i·29-s − 31-s − i·43-s − 1.41·47-s − 1.41i·59-s + 61-s − i·67-s − 1.41·77-s − 1.41·83-s + ⋯
L(s)  = 1  − i·7-s − 1.41i·11-s + i·13-s − 1.41·17-s + 19-s + 1.41·23-s − 1.41i·29-s − 31-s − i·43-s − 1.41·47-s − 1.41i·59-s + 61-s − i·67-s − 1.41·77-s − 1.41·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.151 + 0.988i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.151 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129067687\)
\(L(\frac12)\) \(\approx\) \(1.129067687\)
\(L(1)\) 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 \( 1 \)
good7 \( 1 + iT - T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−8.647195548416301728812867366856, −7.79967175068062478838582834349, −6.97695562599331122924141286857, −6.52723544757181322127181824592, −5.55138330573842782591591755505, −4.68711831587060441023507023964, −3.88206544567739735881686513325, −3.16306431531048493365752755492, −1.95409147217083438530622561938, −0.67145729945342077146109461691, 1.49205303544756842135744264782, 2.53046382973222826594952488985, 3.23958368913712834914507688441, 4.49346416663969505392056860185, 5.14494096548035000142379217667, 5.75030573277880036912123026119, 6.93125320519965745054834139563, 7.20445349872318178010278100031, 8.265499702910278270250644188333, 8.953627717738750769971933881229

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