↑

Properties

Label 2-1200-5.4-c3-0-2
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s + 21.8i·7-s − 9·9-s − 54.6·11-s + 82.7i·13-s + 100. i·17-s − 84.1·19-s − 65.6·21-s + 0.880i·23-s − 27i·27-s + 99.1·29-s + 78.9·31-s − 163. i·33-s − 390. i·37-s − 248.·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.18i·7-s − 0.333·9-s − 1.49·11-s + 1.76i·13-s + 1.43i·17-s − 1.01·19-s − 0.682·21-s + 0.00798i·23-s − 0.192i·27-s + 0.634·29-s + 0.457·31-s − 0.864i·33-s − 1.73i·37-s − 1.01·39-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

\(L(2)\) \(\approx\) \(0.6199887112\)
\(L(\frac12)\) \(\approx\) \(0.6199887112\)
\(L(\frac{5}{2})\) 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 \( 1 - 21.8iT - 343T^{2} \)
11 \( 1 + 54.6T + 1.33e3T^{2} \)
13 \( 1 - 82.7iT - 2.19e3T^{2} \)
17 \( 1 - 100. iT - 4.91e3T^{2} \)
19 \( 1 + 84.1T + 6.85e3T^{2} \)
23 \( 1 - 0.880iT - 1.21e4T^{2} \)
29 \( 1 - 99.1T + 2.43e4T^{2} \)
31 \( 1 - 78.9T + 2.97e4T^{2} \)
37 \( 1 + 390. iT - 5.06e4T^{2} \)
41 \( 1 - 104.T + 6.89e4T^{2} \)
43 \( 1 + 241. iT - 7.95e4T^{2} \)
47 \( 1 - 512. iT - 1.03e5T^{2} \)
53 \( 1 - 284. iT - 1.48e5T^{2} \)
59 \( 1 + 709.T + 2.05e5T^{2} \)
61 \( 1 - 470.T + 2.26e5T^{2} \)
67 \( 1 + 667. iT - 3.00e5T^{2} \)
71 \( 1 - 51.5T + 3.57e5T^{2} \)
73 \( 1 - 371. iT - 3.89e5T^{2} \)
79 \( 1 + 79.3T + 4.93e5T^{2} \)
83 \( 1 - 682. iT - 5.71e5T^{2} \)
89 \( 1 + 628.T + 7.04e5T^{2} \)
97 \( 1 + 1.51e3iT - 9.12e5T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.902088905012267058710428535850, −8.972266840979398592059184430292, −8.560173747502228734951726976792, −7.60315467236424285349090502206, −6.35012694980706314552406352937, −5.77310182717029507612271986173, −4.75201218169182977446895451661, −3.99548480529106354732699163017, −2.64131121467684813148136713613, −1.93517776335600851274284424658, 0.17061687463274343156222542082, 0.910929414765255087092791798109, 2.53037266015840793364535561518, 3.23063613081852013891503192629, 4.65669786622833545140930034541, 5.33976698343496760468432338485, 6.43702572331673070190514754554, 7.31726670968941248285659846701, 7.911811906396956759389390963064, 8.498422359853936172651803114594

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