Properties

Label 2-60e2-5.4-c1-0-25
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  − 4·11-s + 6i·13-s − 6i·17-s − 4·19-s − 2·29-s + 8·31-s + 2i·37-s + 6·41-s − 12i·43-s − 8i·47-s + 7·49-s − 6i·53-s − 12·59-s + 14·61-s + 4i·67-s + ⋯
L(s)  = 1  − 1.20·11-s + 1.66i·13-s − 1.45i·17-s − 0.917·19-s − 0.371·29-s + 1.43·31-s + 0.328i·37-s + 0.937·41-s − 1.82i·43-s − 1.16i·47-s + 49-s − 0.824i·53-s − 1.56·59-s + 1.79·61-s + 0.488i·67-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

\(L(1)\) \(\approx\) \(1.246011035\)
\(L(\frac12)\) \(\approx\) \(1.246011035\)
\(L(\frac{3}{2})\) 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 \( 1 - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 12iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
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

−8.546914401518107204058978036954, −7.57694521132852757032869702172, −6.99634290600883530296573979046, −6.28176010398922427641314122390, −5.27850768104958722113804489999, −4.66299721486102806706325159265, −3.83782041939156324691646444102, −2.64794761222461551270902274564, −2.02409094623636756180326573389, −0.43233629985980295369211591990, 0.960396513874823235630431002852, 2.35982320578521408068811531528, 3.03701392698589328992365024720, 4.08953771212554761910014870521, 4.90529113699697236846665615401, 5.83626622806603151858225137821, 6.19701608648906102378272823166, 7.40295750780192372897750411817, 8.141260748405405423643749122721, 8.301108504865396915362581905499

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