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Properties

Label 2-7200-12.11-c1-0-16
Degree $2$
Conductor $7200$
Sign $0.169 - 0.985i$
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  + 0.864i·7-s − 3.90·11-s + 1.13·13-s − 3.71i·17-s − 1.72i·19-s − 9.03·23-s − 1.26i·29-s − 3.25i·31-s + 6.38·37-s + 6.39i·41-s − 4.77i·43-s + 4.59·47-s + 6.25·49-s + 8.98i·53-s + 8.50·59-s + ⋯
L(s)  = 1  + 0.326i·7-s − 1.17·11-s + 0.314·13-s − 0.900i·17-s − 0.396i·19-s − 1.88·23-s − 0.235i·29-s − 0.584i·31-s + 1.05·37-s + 0.998i·41-s − 0.728i·43-s + 0.670·47-s + 0.893·49-s + 1.23i·53-s + 1.10·59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.169 - 0.985i$
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: \((2,\ 7200,\ (\ :1/2),\ 0.169 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.125529149\)
\(L(\frac12)\) \(\approx\) \(1.125529149\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.864iT - 7T^{2} \)
11 \( 1 + 3.90T + 11T^{2} \)
13 \( 1 - 1.13T + 13T^{2} \)
17 \( 1 + 3.71iT - 17T^{2} \)
19 \( 1 + 1.72iT - 19T^{2} \)
23 \( 1 + 9.03T + 23T^{2} \)
29 \( 1 + 1.26iT - 29T^{2} \)
31 \( 1 + 3.25iT - 31T^{2} \)
37 \( 1 - 6.38T + 37T^{2} \)
41 \( 1 - 6.39iT - 41T^{2} \)
43 \( 1 + 4.77iT - 43T^{2} \)
47 \( 1 - 4.59T + 47T^{2} \)
53 \( 1 - 8.98iT - 53T^{2} \)
59 \( 1 - 8.50T + 59T^{2} \)
61 \( 1 + 9.04T + 61T^{2} \)
67 \( 1 - 11.0iT - 67T^{2} \)
71 \( 1 - 8.10T + 71T^{2} \)
73 \( 1 - 4.47T + 73T^{2} \)
79 \( 1 - 14.2iT - 79T^{2} \)
83 \( 1 + 8.10T + 83T^{2} \)
89 \( 1 + 3.56iT - 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−7.893673773902327827665778537420, −7.62239298259181156635477230102, −6.64837154778759948319529814748, −5.87140761926996933614536556296, −5.38666991750044788886286061533, −4.50282864227069831205173408633, −3.81035581416991624674339286101, −2.64656190035198995920658465942, −2.30594326843680880699275920777, −0.861421334818267400576730665616, 0.32592301824075484628354235872, 1.66337067551559706028322286304, 2.44051011257530473667300898586, 3.48988649074151092735802120851, 4.09249606887239093164550115789, 4.94113924017854139692167209381, 5.79035601793472617053353943263, 6.21400231536352927622477186329, 7.17991709840567110838057301387, 7.891936681753543281107800818486

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