Properties

Label 2-60e2-3.2-c2-0-58
Degree $2$
Conductor $3600$
Sign $-0.577 + 0.816i$
Analytic cond. $98.0928$
Root an. cond. $9.90418$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.48·7-s + 9.17i·11-s − 11.4·13-s + 16.9i·17-s − 26.9·19-s − 4.93i·23-s + 20.5i·29-s − 20.9·31-s + 62.4·37-s + 40.9i·41-s + 1.02·43-s − 86.2i·47-s − 18.8·49-s − 96.0i·53-s − 112. i·59-s + ⋯
L(s)  = 1  + 0.783·7-s + 0.833i·11-s − 0.883·13-s + 0.998i·17-s − 1.41·19-s − 0.214i·23-s + 0.707i·29-s − 0.676·31-s + 1.68·37-s + 0.998i·41-s + 0.0238·43-s − 1.83i·47-s − 0.385·49-s − 1.81i·53-s − 1.90i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.577 + 0.816i$
Analytic conductor: \(98.0928\)
Root analytic conductor: \(9.90418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1),\ -0.577 + 0.816i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5430821839\)
\(L(\frac12)\) \(\approx\) \(0.5430821839\)
\(L(2)\) 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 \( 1 - 5.48T + 49T^{2} \)
11 \( 1 - 9.17iT - 121T^{2} \)
13 \( 1 + 11.4T + 169T^{2} \)
17 \( 1 - 16.9iT - 289T^{2} \)
19 \( 1 + 26.9T + 361T^{2} \)
23 \( 1 + 4.93iT - 529T^{2} \)
29 \( 1 - 20.5iT - 841T^{2} \)
31 \( 1 + 20.9T + 961T^{2} \)
37 \( 1 - 62.4T + 1.36e3T^{2} \)
41 \( 1 - 40.9iT - 1.68e3T^{2} \)
43 \( 1 - 1.02T + 1.84e3T^{2} \)
47 \( 1 + 86.2iT - 2.20e3T^{2} \)
53 \( 1 + 96.0iT - 2.80e3T^{2} \)
59 \( 1 + 112. iT - 3.48e3T^{2} \)
61 \( 1 + 66.9T + 3.72e3T^{2} \)
67 \( 1 + 76T + 4.48e3T^{2} \)
71 \( 1 + 24.0iT - 5.04e3T^{2} \)
73 \( 1 - 18.9T + 5.32e3T^{2} \)
79 \( 1 + 106.T + 6.24e3T^{2} \)
83 \( 1 + 45.1iT - 6.88e3T^{2} \)
89 \( 1 - 115. iT - 7.92e3T^{2} \)
97 \( 1 - 87.0T + 9.40e3T^{2} \)
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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

−8.175355644056295831462056080091, −7.42239964227151466573185473521, −6.68585755190181375170269152474, −5.91151472203802619778613405247, −4.84260700436234128976182806429, −4.50747630967938146846093644902, −3.46587591358928062635254786761, −2.20905949166861389620545552324, −1.69027522279004264595772195596, −0.11787401743627227566642307129, 1.05675586312280959665979588893, 2.26591490869014871557559202262, 2.96337392667676076825959877931, 4.26367909135738346686302967019, 4.67730264170461972576771444745, 5.73828375630187579236408718718, 6.24954397330890988814388813712, 7.52682030430905729377888143914, 7.62878684836462156055211255655, 8.754630229892777687521292889519

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