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Properties

Label 2-2400-24.11-c1-0-50
Degree $2$
Conductor $2400$
Sign $0.612 + 0.790i$
Analytic cond. $19.1640$
Root an. cond. $4.37768$
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  + 1.73i·3-s − 2.99·9-s − 6.92i·17-s + 4·19-s − 8.94·23-s − 5.19i·27-s − 7.74i·31-s − 8.94·47-s + 7·49-s + 11.9·51-s + 4.47·53-s + 6.92i·57-s − 15.4i·61-s − 15.4i·69-s − 7.74i·79-s + ⋯
L(s)  = 1  + 0.999i·3-s − 0.999·9-s − 1.68i·17-s + 0.917·19-s − 1.86·23-s − 0.999i·27-s − 1.39i·31-s − 1.30·47-s + 49-s + 1.68·51-s + 0.614·53-s + 0.917i·57-s − 1.98i·61-s − 1.86i·69-s − 0.871i·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.612 + 0.790i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (2351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.612 + 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.091870824\)
\(L(\frac12)\) \(\approx\) \(1.091870824\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
5 \( 1 \)
good7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 6.92iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 8.94T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 7.74iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 8.94T + 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 15.4iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 7.74iT - 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.065553500223775562541418370866, −8.092152390571993925684056467206, −7.48783039432950710879151652064, −6.37492829890757152923690986950, −5.57188175236577507502216735272, −4.86789457165996237329452454420, −4.02803521959377076695882249305, −3.17528537114985370554770379499, −2.20947453172413770528506591632, −0.37958263831386666004294485887, 1.24886155361514589552201917328, 2.11806964897902627891028492199, 3.25237802149363320592251824163, 4.15150737967276833822002584368, 5.42784428290185321779913141118, 6.00851601602646929213618165406, 6.77683714649729463163547048774, 7.56896994254133447837874540263, 8.276667262366009347868217420889, 8.770028752031780814953370557471

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