Properties

Label 2-60e2-5.4-c1-0-12
Degree $2$
Conductor $3600$
Sign $0.447 - 0.894i$
Analytic cond. $28.7461$
Root an. cond. $5.36154$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·7-s + 6·11-s + 4i·13-s + 6i·17-s − 4·19-s − 6·29-s + 4·31-s + 8i·37-s + 8i·43-s + 3·49-s − 6i·53-s − 6·59-s + 2·61-s + 4i·67-s − 12·71-s + ⋯
L(s)  = 1  − 0.755i·7-s + 1.80·11-s + 1.10i·13-s + 1.45i·17-s − 0.917·19-s − 1.11·29-s + 0.718·31-s + 1.31i·37-s + 1.21i·43-s + 0.428·49-s − 0.824i·53-s − 0.781·59-s + 0.256·61-s + 0.488i·67-s − 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(28.7461\)
Root analytic conductor: \(5.36154\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.775927341\)
\(L(\frac12)\) \(\approx\) \(1.775927341\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.738563574983699868229627773746, −8.007547169977220550418858076983, −7.02058099779801638852536475747, −6.48855197973730516882011188312, −5.97793061647230281331957697206, −4.51360690271049199913910134070, −4.15627070691548670605831293873, −3.41892952375887439061413641771, −1.92425968585657447094574700080, −1.23255323987713508187576117340, 0.56330541303370341959236423617, 1.86056042187846097712020850355, 2.82916427336460765669770946059, 3.74098067043490529809660739880, 4.57249754922739729183021976310, 5.54790154231317012997678937115, 6.09327190919315350626284342643, 6.97124908232947533375272093629, 7.60768448406809336106818724476, 8.631257763503958944764484055077

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