Properties

Label 2-1232-7.2-c1-0-23
Degree $2$
Conductor $1232$
Sign $0.533 + 0.845i$
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  + (−0.746 − 1.29i)3-s + (0.631 − 1.09i)5-s + (2.25 + 1.37i)7-s + (0.385 − 0.667i)9-s + (−0.5 − 0.866i)11-s + 6.62·13-s − 1.88·15-s + (0.512 + 0.888i)17-s + (−1.05 + 1.82i)19-s + (0.0936 − 3.94i)21-s + (−0.923 + 1.60i)23-s + (1.70 + 2.94i)25-s − 5.63·27-s + 4.86·29-s + (0.804 + 1.39i)31-s + ⋯
L(s)  = 1  + (−0.430 − 0.746i)3-s + (0.282 − 0.489i)5-s + (0.853 + 0.520i)7-s + (0.128 − 0.222i)9-s + (−0.150 − 0.261i)11-s + 1.83·13-s − 0.487·15-s + (0.124 + 0.215i)17-s + (−0.241 + 0.417i)19-s + (0.0204 − 0.861i)21-s + (−0.192 + 0.333i)23-s + (0.340 + 0.589i)25-s − 1.08·27-s + 0.903·29-s + (0.144 + 0.250i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $0.533 + 0.845i$
Analytic conductor: \(9.83756\)
Root analytic conductor: \(3.13649\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :1/2),\ 0.533 + 0.845i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.832900928\)
\(L(\frac12)\) \(\approx\) \(1.832900928\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.25 - 1.37i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (0.746 + 1.29i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.631 + 1.09i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 6.62T + 13T^{2} \)
17 \( 1 + (-0.512 - 0.888i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.05 - 1.82i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.923 - 1.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.86T + 29T^{2} \)
31 \( 1 + (-0.804 - 1.39i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.08 + 1.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.49T + 41T^{2} \)
43 \( 1 - 12.3T + 43T^{2} \)
47 \( 1 + (-2.51 + 4.35i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.26 + 2.18i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.77 + 3.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.47 + 2.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.92 + 8.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.86T + 71T^{2} \)
73 \( 1 + (7.61 + 13.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.15 - 7.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.54T + 83T^{2} \)
89 \( 1 + (3.31 - 5.73i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.10T + 97T^{2} \)
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

−9.397813920962261667384519477986, −8.634728232302909732380422591600, −8.091652535304750684983331369766, −7.06141437706680899389547801641, −6.01748465999923177033163217761, −5.70155979867394150400835094366, −4.50826439664054478096276768690, −3.40178587071073102551565292984, −1.80450407424891320991010671652, −1.06697862308045917145829729113, 1.25458792103916249558726660576, 2.66373879834520898789361393920, 4.04377394295757347768152283731, 4.55894391861294736515444390069, 5.61474944408980174077671602803, 6.40978661776023042850439532370, 7.39373593190276029915623093055, 8.276419138422282114141590464992, 9.035868185791206165795834351965, 10.27659965886258945854017098277

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