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Properties

Label 2-1200-240.107-c0-0-1
Degree $2$
Conductor $1200$
Sign $0.584 + 0.811i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − i·3-s + 4-s − i·6-s + 8-s − 9-s − i·12-s + 16-s + (1 − i)17-s − 18-s + (−1 + i)19-s + (−1 − i)23-s − i·24-s + i·27-s + 2i·31-s + 32-s + ⋯
L(s)  = 1  + 2-s − i·3-s + 4-s − i·6-s + 8-s − 9-s − i·12-s + 16-s + (1 − i)17-s − 18-s + (−1 + i)19-s + (−1 − i)23-s − i·24-s + i·27-s + 2i·31-s + 32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.584 + 0.811i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ 0.584 + 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.910556555\)
\(L(\frac12)\) \(\approx\) \(1.910556555\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-1 + i)T - iT^{2} \)
19 \( 1 + (1 - i)T - iT^{2} \)
23 \( 1 + (1 + i)T + iT^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1 + i)T + iT^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (1 - i)T - iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.13611474036246154783996907708, −8.718697627778569301825632389079, −7.961204125518065708202594658870, −7.17695853182858015602995782105, −6.41779075784429779804635795656, −5.70018132663773805848785583327, −4.78289035010519154174425609022, −3.56545086274853630540864278132, −2.61251717099937078186702596128, −1.51535550607365298497186594261, 2.06674253838762747478721304463, 3.26610426994334539109150757663, 4.03576538935355803687547720536, 4.80249613123877603875270769552, 5.80543424145095602332606157655, 6.30469639710358608934824143621, 7.65193474867595104056893578937, 8.310671124963649093714141099915, 9.546246945455479574045541639142, 10.10929004588547097289150736967

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