Properties

Label 2-7200-40.29-c1-0-21
Degree $2$
Conductor $7200$
Sign $-0.606 - 0.794i$
Analytic cond. $57.4922$
Root an. cond. $7.58236$
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  + 0.941i·7-s + 4.49i·11-s + 5.55·13-s + 7.55i·17-s + 1.05i·19-s − 1.05i·23-s − 2i·29-s − 3.55·31-s − 7.43·37-s + 3.88·41-s − 1.88·43-s + 10.0i·47-s + 6.11·49-s + 2·53-s + 8.49i·59-s + ⋯
L(s)  = 1  + 0.355i·7-s + 1.35i·11-s + 1.54·13-s + 1.83i·17-s + 0.242i·19-s − 0.220i·23-s − 0.371i·29-s − 0.638·31-s − 1.22·37-s + 0.606·41-s − 0.287·43-s + 1.46i·47-s + 0.873·49-s + 0.274·53-s + 1.10i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7200\)    =    \(2^{5} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.606 - 0.794i$
Analytic conductor: \(57.4922\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7200} (2449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7200,\ (\ :1/2),\ -0.606 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.658611429\)
\(L(\frac12)\) \(\approx\) \(1.658611429\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 - 0.941iT - 7T^{2} \)
11 \( 1 - 4.49iT - 11T^{2} \)
13 \( 1 - 5.55T + 13T^{2} \)
17 \( 1 - 7.55iT - 17T^{2} \)
19 \( 1 - 1.05iT - 19T^{2} \)
23 \( 1 + 1.05iT - 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 3.55T + 31T^{2} \)
37 \( 1 + 7.43T + 37T^{2} \)
41 \( 1 - 3.88T + 41T^{2} \)
43 \( 1 + 1.88T + 43T^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 - 8.49iT - 59T^{2} \)
61 \( 1 - 8.99iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + 5.88T + 83T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 + 17.1iT - 97T^{2} \)
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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

−8.272130163809994099881230406813, −7.47159245350437495102394288831, −6.76313454340726471582232410841, −5.95701166566856379965483723735, −5.60656689193088464966477818137, −4.35411347171690267095932900755, −4.03290502182317811519387618110, −3.05553210248927935592162162498, −1.94138792493240170770397333560, −1.37183320681353736341220413611, 0.42129723274696451594403581946, 1.24702037831050993266105718060, 2.49393296027004046784090276515, 3.47008314354297202701059942539, 3.77334784152798513174311838691, 5.03142027362951578300189955886, 5.48266807233924748915033714101, 6.35096942335990621196539268696, 6.93182417435730753306413979014, 7.66907194145163925418231092210

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