↑

Properties

Label 2-1176-8.5-c1-0-81
Degree $2$
Conductor $1176$
Sign $-0.992 + 0.121i$
Analytic cond. $9.39040$
Root an. cond. $3.06437$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.19 − 0.756i)2-s − i·3-s + (0.856 − 1.80i)4-s − 4.10i·5-s + (−0.756 − 1.19i)6-s + (−0.343 − 2.80i)8-s − 9-s + (−3.10 − 4.90i)10-s − 2.67i·11-s + (−1.80 − 0.856i)12-s + 3.02i·13-s − 4.10·15-s + (−2.53 − 3.09i)16-s + 5.12·17-s + (−1.19 + 0.756i)18-s + 2.78i·19-s + ⋯
L(s)  = 1  + (0.845 − 0.534i)2-s − 0.577i·3-s + (0.428 − 0.903i)4-s − 1.83i·5-s + (−0.308 − 0.487i)6-s + (−0.121 − 0.992i)8-s − 0.333·9-s + (−0.981 − 1.55i)10-s − 0.807i·11-s + (−0.521 − 0.247i)12-s + 0.838i·13-s − 1.05·15-s + (−0.633 − 0.773i)16-s + 1.24·17-s + (−0.281 + 0.178i)18-s + 0.637i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.121i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.992 + 0.121i$
Analytic conductor: \(9.39040\)
Root analytic conductor: \(3.06437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1176,\ (\ :1/2),\ -0.992 + 0.121i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.644511934\)
\(L(\frac12)\) \(\approx\) \(2.644511934\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.756i)T \)
3 \( 1 + iT \)
7 \( 1 \)
good5 \( 1 + 4.10iT - 5T^{2} \)
11 \( 1 + 2.67iT - 11T^{2} \)
13 \( 1 - 3.02iT - 13T^{2} \)
17 \( 1 - 5.12T + 17T^{2} \)
19 \( 1 - 2.78iT - 19T^{2} \)
23 \( 1 - 7.12T + 23T^{2} \)
29 \( 1 - 8.83iT - 29T^{2} \)
31 \( 1 - 1.42T + 31T^{2} \)
37 \( 1 + 1.42iT - 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
43 \( 1 + 2.39iT - 43T^{2} \)
47 \( 1 - 9.56T + 47T^{2} \)
53 \( 1 - 2.78iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 5.17iT - 61T^{2} \)
67 \( 1 - 0.244iT - 67T^{2} \)
71 \( 1 + 4.27T + 71T^{2} \)
73 \( 1 - 4.15T + 73T^{2} \)
79 \( 1 - 6.25T + 79T^{2} \)
83 \( 1 + 9.35iT - 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + 6.69T + 97T^{2} \)
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

−9.194829296315927150552915819129, −8.796535599054040495365152779270, −7.74023032615068997251296981503, −6.70321857124580372206158194967, −5.57592039105291370812231199585, −5.22199243640718329414052321314, −4.16730612445960539540793907595, −3.15987746437749513216674021682, −1.61884589458139723971548233748, −0.928406191025036971333334412505, 2.53646152289368337652540636060, 3.11346337696149302092315294531, 4.01198626763122450767780633303, 5.12875518441308893020580274596, 5.95390661410200643898433749789, 6.83975682543037016025991740192, 7.43123543810503189603388172401, 8.193847555186321381569450683843, 9.565191944757627222771529434335, 10.27240159676167193825868927195

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