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Properties

Label 2-7200-5.4-c1-0-87
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  − 6i·13-s − 2i·17-s − 10·29-s − 2i·37-s − 10·41-s + 7·49-s + 14i·53-s − 10·61-s + 6i·73-s + 10·89-s + 18i·97-s + 2·101-s − 6·109-s − 14i·113-s + ⋯
L(s)  = 1  − 1.66i·13-s − 0.485i·17-s − 1.85·29-s − 0.328i·37-s − 1.56·41-s + 49-s + 1.92i·53-s − 1.28·61-s + 0.702i·73-s + 1.05·89-s + 1.82i·97-s + 0.199·101-s − 0.574·109-s − 1.31i·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: \(1\)
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 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
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 \( 1 - 10T + 89T^{2} \)
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

−7.58105784290346766678423619659, −6.96051038883707895771903179754, −5.93546962492472763118077016525, −5.51106722177441903636684142364, −4.76656821210134003325216795712, −3.78012924740910852159521057776, −3.12272234514041722068961140726, −2.27519739123717602761760911783, −1.12805719568410342545126853763, 0, 1.58198389630244349639762946872, 2.11617146244461882434494004758, 3.35724843828204834226570607259, 3.98309358878162299332738758755, 4.74540745529349042005529653558, 5.51725418449525689823414010425, 6.33848680016897866159295652329, 6.89519821979578037628632491239, 7.56108930822282178654069748857

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