Properties

Label 2-2400-5.4-c1-0-10
Degree $2$
Conductor $2400$
Sign $0.894 - 0.447i$
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  i·3-s + i·7-s − 9-s − 4·11-s + 3i·13-s − 4i·17-s + 19-s + 21-s + i·27-s + 8·29-s + 31-s + 4i·33-s + 2i·37-s + 3·39-s + 2·41-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.20·11-s + 0.832i·13-s − 0.970i·17-s + 0.229·19-s + 0.218·21-s + 0.192i·27-s + 1.48·29-s + 0.179·31-s + 0.696i·33-s + 0.328i·37-s + 0.480·39-s + 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.435124514\)
\(L(\frac12)\) \(\approx\) \(1.435124514\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 11iT - 43T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 - 11T + 61T^{2} \)
67 \( 1 - 9iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 2iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 11iT - 97T^{2} \)
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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

−8.863771024278024627842282771767, −8.260092773179904227575010232715, −7.43352426691261945998355987388, −6.81525462667645190425728318182, −5.91904881863551784141706248904, −5.13555495728022412626978133650, −4.33400306114065246649581609742, −2.91565845745533697559344947051, −2.39314886131597067972497068546, −0.998130716069444828159227634325, 0.58503994907227031916759891316, 2.23442900038762032431635233913, 3.20362783544027521640094087752, 4.03528920200095265461536978678, 5.02606133232270442284528364130, 5.59003678178637576274541317285, 6.54209096557011535309008619499, 7.52265549617193417269822822219, 8.201511441752691545701327914464, 8.780724800611461752648206592333

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