Properties

Label 2-1224-17.16-c3-0-47
Degree $2$
Conductor $1224$
Sign $0.782 - 0.622i$
Analytic cond. $72.2183$
Root an. cond. $8.49813$
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  + 11.5i·5-s + 23.0i·7-s − 1.19i·11-s + 58.3·13-s + (54.8 − 43.6i)17-s + 138.·19-s − 180. i·23-s − 7.98·25-s − 115. i·29-s − 210. i·31-s − 265.·35-s − 210. i·37-s + 297. i·41-s + 174.·43-s + 199.·47-s + ⋯
L(s)  = 1  + 1.03i·5-s + 1.24i·7-s − 0.0327i·11-s + 1.24·13-s + (0.782 − 0.622i)17-s + 1.67·19-s − 1.64i·23-s − 0.0638·25-s − 0.736i·29-s − 1.22i·31-s − 1.28·35-s − 0.936i·37-s + 1.13i·41-s + 0.619·43-s + 0.619·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.782 - 0.622i$
Analytic conductor: \(72.2183\)
Root analytic conductor: \(8.49813\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :3/2),\ 0.782 - 0.622i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.627151602\)
\(L(\frac12)\) \(\approx\) \(2.627151602\)
\(L(\frac{5}{2})\) 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 \( 1 \)
17 \( 1 + (-54.8 + 43.6i)T \)
good5 \( 1 - 11.5iT - 125T^{2} \)
7 \( 1 - 23.0iT - 343T^{2} \)
11 \( 1 + 1.19iT - 1.33e3T^{2} \)
13 \( 1 - 58.3T + 2.19e3T^{2} \)
19 \( 1 - 138.T + 6.85e3T^{2} \)
23 \( 1 + 180. iT - 1.21e4T^{2} \)
29 \( 1 + 115. iT - 2.43e4T^{2} \)
31 \( 1 + 210. iT - 2.97e4T^{2} \)
37 \( 1 + 210. iT - 5.06e4T^{2} \)
41 \( 1 - 297. iT - 6.89e4T^{2} \)
43 \( 1 - 174.T + 7.95e4T^{2} \)
47 \( 1 - 199.T + 1.03e5T^{2} \)
53 \( 1 - 706.T + 1.48e5T^{2} \)
59 \( 1 - 182.T + 2.05e5T^{2} \)
61 \( 1 - 489. iT - 2.26e5T^{2} \)
67 \( 1 + 167.T + 3.00e5T^{2} \)
71 \( 1 - 818. iT - 3.57e5T^{2} \)
73 \( 1 + 1.09e3iT - 3.89e5T^{2} \)
79 \( 1 + 110. iT - 4.93e5T^{2} \)
83 \( 1 + 118.T + 5.71e5T^{2} \)
89 \( 1 + 400.T + 7.04e5T^{2} \)
97 \( 1 - 924. iT - 9.12e5T^{2} \)
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

−9.423864259930650485542204253219, −8.667272167174874226302762067854, −7.79032891960988645613170409501, −6.93745727969605041633160027467, −5.96940611841168790844624814355, −5.52196803437913045541751117751, −4.13038366588530833994395340708, −3.00856938939188668875190301774, −2.44069226292818984970349821845, −0.862789063891583828395430442126, 1.02775468068570632538468626041, 1.30064357202631963396373009876, 3.37712952458858604108434571938, 3.89891505956146213982075263716, 5.10637153719649900276980449015, 5.68383413319315299463644465709, 6.96669639382415972096522784555, 7.61424491837691362980203735914, 8.485653013725259611381357039146, 9.204019810822036482857348123968

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