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Properties

Label 2-2400-120.83-c0-0-2
Degree $2$
Conductor $2400$
Sign $0.997 - 0.0706i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
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  + (0.965 + 0.258i)3-s + (0.866 + 0.499i)9-s − 1.73i·11-s + (0.707 + 0.707i)17-s + i·19-s + (0.707 + 0.707i)27-s + (0.448 − 1.67i)33-s − 1.73i·41-s − i·49-s + (0.500 + 0.866i)51-s + (−0.258 + 0.965i)57-s + (−1.22 + 1.22i)67-s + (1.22 + 1.22i)73-s + (0.500 + 0.866i)81-s + (−0.707 + 0.707i)83-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)3-s + (0.866 + 0.499i)9-s − 1.73i·11-s + (0.707 + 0.707i)17-s + i·19-s + (0.707 + 0.707i)27-s + (0.448 − 1.67i)33-s − 1.73i·41-s − i·49-s + (0.500 + 0.866i)51-s + (−0.258 + 0.965i)57-s + (−1.22 + 1.22i)67-s + (1.22 + 1.22i)73-s + (0.500 + 0.866i)81-s + (−0.707 + 0.707i)83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0706i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.997 - 0.0706i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :0),\ 0.997 - 0.0706i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.740982875\)
\(L(\frac12)\) \(\approx\) \(1.740982875\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 1.73iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + 1.73T + T^{2} \)
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

−8.940608650598639242550013943080, −8.428951379803637082200243264904, −7.899927691290909960991441917795, −6.97784999574624806580991221004, −5.93169416825047921947711155364, −5.32782664637918805179595172355, −3.95262560438399134076474247375, −3.55139343666963079446374965515, −2.57505236827649852067124662268, −1.34080817360016857976838876563, 1.42053107648604393934414043739, 2.44715934206757391219450084438, 3.22202114026809409496521391928, 4.41716709423365387221303278143, 4.89687937249419569653508893622, 6.24586125274362133903378517055, 7.12026129537381863342796840192, 7.52115232405165563315131113298, 8.309145458012088272359262970015, 9.343835183422016298299358924198

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