Properties

Label 2-1200-5.4-c3-0-12
Degree $2$
Conductor $1200$
Sign $0.894 - 0.447i$
Analytic cond. $70.8022$
Root an. cond. $8.41440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s − 24i·7-s − 9·9-s − 52·11-s + 22i·13-s + 14i·17-s − 20·19-s − 72·21-s + 168i·23-s + 27i·27-s − 230·29-s + 288·31-s + 156i·33-s + 34i·37-s + 66·39-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.29i·7-s − 0.333·9-s − 1.42·11-s + 0.469i·13-s + 0.199i·17-s − 0.241·19-s − 0.748·21-s + 1.52i·23-s + 0.192i·27-s − 1.47·29-s + 1.66·31-s + 0.822i·33-s + 0.151i·37-s + 0.270·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(70.8022\)
Root analytic conductor: \(8.41440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.171041577\)
\(L(\frac12)\) \(\approx\) \(1.171041577\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
5 \( 1 \)
good7 \( 1 + 24iT - 343T^{2} \)
11 \( 1 + 52T + 1.33e3T^{2} \)
13 \( 1 - 22iT - 2.19e3T^{2} \)
17 \( 1 - 14iT - 4.91e3T^{2} \)
19 \( 1 + 20T + 6.85e3T^{2} \)
23 \( 1 - 168iT - 1.21e4T^{2} \)
29 \( 1 + 230T + 2.43e4T^{2} \)
31 \( 1 - 288T + 2.97e4T^{2} \)
37 \( 1 - 34iT - 5.06e4T^{2} \)
41 \( 1 - 122T + 6.89e4T^{2} \)
43 \( 1 - 188iT - 7.95e4T^{2} \)
47 \( 1 - 256iT - 1.03e5T^{2} \)
53 \( 1 + 338iT - 1.48e5T^{2} \)
59 \( 1 - 100T + 2.05e5T^{2} \)
61 \( 1 - 742T + 2.26e5T^{2} \)
67 \( 1 + 84iT - 3.00e5T^{2} \)
71 \( 1 - 328T + 3.57e5T^{2} \)
73 \( 1 + 38iT - 3.89e5T^{2} \)
79 \( 1 + 240T + 4.93e5T^{2} \)
83 \( 1 + 1.21e3iT - 5.71e5T^{2} \)
89 \( 1 + 330T + 7.04e5T^{2} \)
97 \( 1 + 866iT - 9.12e5T^{2} \)
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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

−9.584102902399696338862671948576, −8.368261792575798144698269938489, −7.66087731357082742310527842421, −7.17261689491423870930504253545, −6.16942378039672201968295999020, −5.21621635503340887771324000148, −4.21508581911074986114216732787, −3.19720729963833811024921012866, −1.99882877623340155393399687571, −0.840521063463406905970192718478, 0.34723084631522976659882575486, 2.32942035284947921377474169413, 2.82584000500249093327824900980, 4.17439512547816117632999192290, 5.24526577752151801643046017886, 5.64059701560524747636056617506, 6.73126291108450120801948082546, 7.978729162394650041315246579078, 8.471467123230260455410486401530, 9.305199251133958269572880601264

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