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Properties

Label 2-7200-12.11-c1-0-39
Degree $2$
Conductor $7200$
Sign $0.985 - 0.169i$
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  + 2.41i·7-s − 5.41·11-s + 6.65·13-s − 3.41i·17-s + 7.24i·19-s + 2.58·23-s − 10.2i·29-s − 5.24i·31-s + 6.82·37-s − 4.82i·41-s − 3.58i·43-s − 7.41·47-s + 1.17·49-s + 0.828i·53-s + 5.07·59-s + ⋯
L(s)  = 1  + 0.912i·7-s − 1.63·11-s + 1.84·13-s − 0.828i·17-s + 1.66i·19-s + 0.539·23-s − 1.90i·29-s − 0.941i·31-s + 1.12·37-s − 0.754i·41-s − 0.546i·43-s − 1.08·47-s + 0.167·49-s + 0.113i·53-s + 0.660·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ 0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.892946591\)
\(L(\frac12)\) \(\approx\) \(1.892946591\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41iT - 7T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 - 6.65T + 13T^{2} \)
17 \( 1 + 3.41iT - 17T^{2} \)
19 \( 1 - 7.24iT - 19T^{2} \)
23 \( 1 - 2.58T + 23T^{2} \)
29 \( 1 + 10.2iT - 29T^{2} \)
31 \( 1 + 5.24iT - 31T^{2} \)
37 \( 1 - 6.82T + 37T^{2} \)
41 \( 1 + 4.82iT - 41T^{2} \)
43 \( 1 + 3.58iT - 43T^{2} \)
47 \( 1 + 7.41T + 47T^{2} \)
53 \( 1 - 0.828iT - 53T^{2} \)
59 \( 1 - 5.07T + 59T^{2} \)
61 \( 1 + 1.82T + 61T^{2} \)
67 \( 1 + 3.24iT - 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 16.4T + 73T^{2} \)
79 \( 1 - 12iT - 79T^{2} \)
83 \( 1 + 4.58T + 83T^{2} \)
89 \( 1 - 12iT - 89T^{2} \)
97 \( 1 - 9T + 97T^{2} \)
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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

−8.076415248459085086061857466111, −7.43650191707526471660655255000, −6.29104517983576202147799798215, −5.81060562612558195496032351338, −5.36198875534597758902456013450, −4.33059145868896031290073462302, −3.53887867411739176625892938354, −2.66538923387610898329670939853, −1.98450775448176746071945182712, −0.67920104162489992579883489351, 0.73391924644291854916889066924, 1.61379356915973691451341248411, 2.93660792828993910499531775638, 3.36792562242174965657467151721, 4.43226791724868203331082988113, 4.98349856849110734636748541898, 5.82343223346684487944331528592, 6.59369036203547291276378785460, 7.19366801281770301406378407084, 7.908965359466818797667200393683

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