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Properties

Label 2-7200-5.4-c1-0-86
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·13-s − 6i·17-s − 4·19-s − 8i·23-s − 2·29-s − 4·31-s + 10i·37-s − 2·41-s − 4i·43-s + 8i·47-s + 7·49-s − 2i·53-s − 8·59-s − 2·61-s + 12i·67-s + ⋯
L(s)  = 1  − 0.554i·13-s − 1.45i·17-s − 0.917·19-s − 1.66i·23-s − 0.371·29-s − 0.718·31-s + 1.64i·37-s − 0.312·41-s − 0.609i·43-s + 1.16i·47-s + 49-s − 0.274i·53-s − 1.04·59-s − 0.256·61-s + 1.46i·67-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: \(1\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
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

−7.45597293214862485019806817641, −6.85979326186194107822462303097, −6.18478919183988991642986335142, −5.35289136676602323865334265258, −4.69267083886377790670455083323, −3.97803985225026872917378679028, −2.91621823816819662221628145483, −2.38857557611843720145244641424, −1.10703792391520391895818679877, 0, 1.57846482229399252398192240085, 2.10747171846997445518957410308, 3.41541706144279670535189878438, 3.91704313455398334824026632536, 4.71818617708420005636477493361, 5.72203434450821139981966836278, 6.06857206511898992803407515912, 7.02917218567925270225371546353, 7.56820718174043628869624118596

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