↑

Properties

Label 2-1232-7.4-c1-0-8
Degree $2$
Conductor $1232$
Sign $0.911 - 0.411i$
Analytic cond. $9.83756$
Root an. cond. $3.13649$
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  + (−0.794 + 1.37i)3-s + (−1.64 − 2.84i)5-s + (−2.64 − 0.0963i)7-s + (0.238 + 0.413i)9-s + (−0.5 + 0.866i)11-s − 4.98·13-s + 5.22·15-s + (1.84 − 3.20i)17-s + (2.84 + 4.93i)19-s + (2.23 − 3.56i)21-s + (0.349 + 0.605i)23-s + (−2.90 + 5.03i)25-s − 5.52·27-s + 7.68·29-s + (5.25 − 9.10i)31-s + ⋯
L(s)  = 1  + (−0.458 + 0.794i)3-s + (−0.735 − 1.27i)5-s + (−0.999 − 0.0364i)7-s + (0.0795 + 0.137i)9-s + (−0.150 + 0.261i)11-s − 1.38·13-s + 1.34·15-s + (0.448 − 0.777i)17-s + (0.653 + 1.13i)19-s + (0.487 − 0.776i)21-s + (0.0729 + 0.126i)23-s + (−0.581 + 1.00i)25-s − 1.06·27-s + 1.42·29-s + (0.943 − 1.63i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8753560170\)
\(L(\frac12)\) \(\approx\) \(0.8753560170\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (2.64 + 0.0963i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.794 - 1.37i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.64 + 2.84i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 4.98T + 13T^{2} \)
17 \( 1 + (-1.84 + 3.20i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.84 - 4.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.349 - 0.605i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.68T + 29T^{2} \)
31 \( 1 + (-5.25 + 9.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.28 - 9.15i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.81T + 41T^{2} \)
43 \( 1 + 1.63T + 43T^{2} \)
47 \( 1 + (-1.15 - 1.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.60 + 2.78i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.88 + 6.72i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.47 + 4.29i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.810 - 1.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.30T + 71T^{2} \)
73 \( 1 + (2.70 - 4.68i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.36 + 2.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.65T + 83T^{2} \)
89 \( 1 + (-6.43 - 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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

−9.868446572123935307325910463173, −9.253570586778946258594517263578, −7.988224858653817578425739087784, −7.58649858032278560772069427542, −6.29977720678673086126022701267, −5.22815406380536683050186376608, −4.69977337626448632543905973978, −3.93609528969249748849854885758, −2.68020414429308571150446293530, −0.74215947158386420879381515431, 0.64550114170880522297047788687, 2.60690691233471318663579397865, 3.21356984783010834134914356705, 4.41286158516751917019911100796, 5.76451260141269632901253424475, 6.56177690982101397263343654866, 7.15933696925842595935862005810, 7.57454525538851205920018166353, 8.831611774005310339345171173393, 9.866663006425903672140014474153

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