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Properties

Label 2-1200-5.4-c3-0-16
Degree $2$
Conductor $1200$
Sign $-0.447 - 0.894i$
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 + 7i·7-s − 9·9-s + 54·11-s + 55i·13-s + 18i·17-s − 25·19-s − 21·21-s − 18i·23-s − 27i·27-s + 54·29-s + 271·31-s + 162i·33-s + 314i·37-s − 165·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.377i·7-s − 0.333·9-s + 1.48·11-s + 1.17i·13-s + 0.256i·17-s − 0.301·19-s − 0.218·21-s − 0.163i·23-s − 0.192i·27-s + 0.345·29-s + 1.57·31-s + 0.854i·33-s + 1.39i·37-s − 0.677·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.447 - 0.894i$
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: \((2,\ 1200,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.017376315\)
\(L(\frac12)\) \(\approx\) \(2.017376315\)
\(L(\frac{5}{2})\) 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 \( 1 - 7iT - 343T^{2} \)
11 \( 1 - 54T + 1.33e3T^{2} \)
13 \( 1 - 55iT - 2.19e3T^{2} \)
17 \( 1 - 18iT - 4.91e3T^{2} \)
19 \( 1 + 25T + 6.85e3T^{2} \)
23 \( 1 + 18iT - 1.21e4T^{2} \)
29 \( 1 - 54T + 2.43e4T^{2} \)
31 \( 1 - 271T + 2.97e4T^{2} \)
37 \( 1 - 314iT - 5.06e4T^{2} \)
41 \( 1 + 360T + 6.89e4T^{2} \)
43 \( 1 + 163iT - 7.95e4T^{2} \)
47 \( 1 + 522iT - 1.03e5T^{2} \)
53 \( 1 - 36iT - 1.48e5T^{2} \)
59 \( 1 - 126T + 2.05e5T^{2} \)
61 \( 1 - 47T + 2.26e5T^{2} \)
67 \( 1 - 343iT - 3.00e5T^{2} \)
71 \( 1 - 1.08e3T + 3.57e5T^{2} \)
73 \( 1 - 1.05e3iT - 3.89e5T^{2} \)
79 \( 1 + 568T + 4.93e5T^{2} \)
83 \( 1 - 1.42e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.44e3T + 7.04e5T^{2} \)
97 \( 1 + 439iT - 9.12e5T^{2} \)
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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

−9.704153871904488121429806842047, −8.721794412025526577769542282224, −8.424326200094274154299585098569, −6.81318786416026731162709429189, −6.51104638832887127008378911074, −5.32049161632393736882652333550, −4.34482991094341921657101712414, −3.70631398055409540235275342180, −2.40772034300162293554848723998, −1.21639446469872368141758830268, 0.52899772828367240462423674364, 1.44944424887702306612152415128, 2.78148908002658107576742104953, 3.78049889278053770911545713004, 4.79937358438954831285970096414, 5.95482858596549819677102579659, 6.59644822228975531186182986916, 7.45582428244387797958789774062, 8.227156027655877752073021621797, 9.043489549090572496222499366155

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