Properties

Label 2-7200-5.4-c1-0-82
Degree $2$
Conductor $7200$
Sign $-0.894 + 0.447i$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
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  − 6i·13-s − 8i·17-s + 4·29-s + 2i·37-s − 8·41-s + 7·49-s − 4i·53-s − 10·61-s + 6i·73-s − 16·89-s + 18i·97-s − 20·101-s + 6·109-s + 16i·113-s + ⋯
L(s)  = 1  − 1.66i·13-s − 1.94i·17-s + 0.742·29-s + 0.328i·37-s − 1.24·41-s + 49-s − 0.549i·53-s − 1.28·61-s + 0.702i·73-s − 1.69·89-s + 1.82i·97-s − 1.99·101-s + 0.574·109-s + 1.50i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (6049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.028836285\)
\(L(\frac12)\) \(\approx\) \(1.028836285\)
\(L(\frac{3}{2})\) 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 \( 1 + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 8iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 - 18iT - 97T^{2} \)
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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

−7.63258227351985697197226104506, −7.00249500852540546246282925213, −6.26238532346967097909365385571, −5.26262743885426819639135053220, −5.07693297233372568815288257069, −3.98004333308697781755675958262, −3.01080352445895198015084190465, −2.61075591374592867394623957953, −1.20164235864958435148125206727, −0.25545635836240195624955011582, 1.40895280132758086124764977319, 2.02524808946063722360484240027, 3.13071149525068263688794427413, 4.08200240507964867847410175023, 4.43543629726756709820176276281, 5.49627882217037454259614993252, 6.23073535521833880121412719357, 6.74314218219802026979647005848, 7.47575203211756888208854712107, 8.380300118131861101335132286099

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