Properties

Label 2-1200-3.2-c2-0-52
Degree $2$
Conductor $1200$
Sign $i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 9·9-s + 14i·17-s − 22·19-s − 34i·23-s − 27i·27-s − 2·31-s − 14i·47-s − 49·49-s − 42·51-s − 86i·53-s − 66i·57-s − 118·61-s + 102·69-s + 98·79-s + ⋯
L(s)  = 1  + i·3-s − 9-s + 0.823i·17-s − 1.15·19-s − 1.47i·23-s i·27-s − 0.0645·31-s − 0.297i·47-s − 0.999·49-s − 0.823·51-s − 1.62i·53-s − 1.15i·57-s − 1.93·61-s + 1.47·69-s + 1.24·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5023341546\)
\(L(\frac12)\) \(\approx\) \(0.5023341546\)
\(L(2)\) 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 \( 1 - 3iT \)
5 \( 1 \)
good7 \( 1 + 49T^{2} \)
11 \( 1 - 121T^{2} \)
13 \( 1 + 169T^{2} \)
17 \( 1 - 14iT - 289T^{2} \)
19 \( 1 + 22T + 361T^{2} \)
23 \( 1 + 34iT - 529T^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + 2T + 961T^{2} \)
37 \( 1 + 1.36e3T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 + 14iT - 2.20e3T^{2} \)
53 \( 1 + 86iT - 2.80e3T^{2} \)
59 \( 1 - 3.48e3T^{2} \)
61 \( 1 + 118T + 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 5.32e3T^{2} \)
79 \( 1 - 98T + 6.24e3T^{2} \)
83 \( 1 + 154iT - 6.88e3T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + 9.40e3T^{2} \)
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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

−9.358092223404747095507577888221, −8.591859311204909485854456928467, −8.021402629648115208490292992016, −6.63826214934405337576464961003, −5.99276270125465020971921798772, −4.87882451913006771059841710368, −4.20691802808942260616970166799, −3.23092271652236388101507465236, −2.06061834793225165097105817816, −0.14864680354485370019216411565, 1.27393243802652374639847685657, 2.37638601668693556262988591472, 3.41379623888030203423208867701, 4.71604365003280570270281432213, 5.72900622309570724465978641752, 6.48293632829349111989369110147, 7.35600749831495715759144378672, 7.956787023903309235597167625612, 8.902953340774729244888414740141, 9.579636158406318047917841490552

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