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Properties

Label 2-2400-12.11-c1-0-15
Degree $2$
Conductor $2400$
Sign $-0.558 - 0.829i$
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  + (−0.332 + 1.69i)3-s − 1.60i·7-s + (−2.77 − 1.13i)9-s − 1.66·11-s + 2.64·13-s + 4.67i·17-s − 5.55i·19-s + (2.71 + 0.532i)21-s + 2.16·23-s + (2.84 − 4.34i)27-s + 5.40i·29-s + 7.43i·31-s + (0.553 − 2.82i)33-s − 6.24·37-s + (−0.880 + 4.49i)39-s + ⋯
L(s)  = 1  + (−0.192 + 0.981i)3-s − 0.604i·7-s + (−0.926 − 0.377i)9-s − 0.501·11-s + 0.733·13-s + 1.13i·17-s − 1.27i·19-s + (0.593 + 0.116i)21-s + 0.450·23-s + (0.548 − 0.836i)27-s + 1.00i·29-s + 1.33i·31-s + (0.0964 − 0.492i)33-s − 1.02·37-s + (−0.141 + 0.720i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $-0.558 - 0.829i$
Analytic conductor: \(19.1640\)
Root analytic conductor: \(4.37768\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :1/2),\ -0.558 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.148712954\)
\(L(\frac12)\) \(\approx\) \(1.148712954\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.332 - 1.69i)T \)
5 \( 1 \)
good7 \( 1 + 1.60iT - 7T^{2} \)
11 \( 1 + 1.66T + 11T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 - 4.67iT - 17T^{2} \)
19 \( 1 + 5.55iT - 19T^{2} \)
23 \( 1 - 2.16T + 23T^{2} \)
29 \( 1 - 5.40iT - 29T^{2} \)
31 \( 1 - 7.43iT - 31T^{2} \)
37 \( 1 + 6.24T + 37T^{2} \)
41 \( 1 - 11.8iT - 41T^{2} \)
43 \( 1 + 3.39iT - 43T^{2} \)
47 \( 1 - 8.52T + 47T^{2} \)
53 \( 1 - 8.83iT - 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 3.67T + 61T^{2} \)
67 \( 1 - 5.49iT - 67T^{2} \)
71 \( 1 + 15.9T + 71T^{2} \)
73 \( 1 - 3.59T + 73T^{2} \)
79 \( 1 - 9.55iT - 79T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 - 3.32iT - 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−8.964813434980891663351211544666, −8.837681005740094795864937020158, −7.74985292589565023936654762173, −6.83387662661520404230018745122, −6.05109532604361242034718440292, −5.14111266962657648460390033914, −4.47991320801096958219025653616, −3.58607268903871563787642399571, −2.83117678148938444764174964712, −1.20598224711074772344134856817, 0.42878410811763351750187699574, 1.80516983352089947497862687412, 2.63147076012695814964483969885, 3.66997499255708714348926184937, 4.95133698601721852772585317968, 5.79044103829444565502846189545, 6.20640827372703013937019238694, 7.34787222714900215905504346608, 7.75538044932218799501203205044, 8.660425282598383161556828915272

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