↑

Properties

Label 2-126-63.59-c1-0-1
Degree $2$
Conductor $126$
Sign $0.786 + 0.617i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
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  + (−0.866 − 0.5i)2-s + (−1.72 + 0.148i)3-s + (0.499 + 0.866i)4-s + 0.967·5-s + (1.56 + 0.734i)6-s + (2.40 − 1.11i)7-s − 0.999i·8-s + (2.95 − 0.511i)9-s + (−0.837 − 0.483i)10-s − 5.57i·11-s + (−0.991 − 1.42i)12-s + (3.76 + 2.17i)13-s + (−2.63 − 0.238i)14-s + (−1.66 + 0.143i)15-s + (−0.5 + 0.866i)16-s + (−1.97 + 3.41i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.996 + 0.0855i)3-s + (0.249 + 0.433i)4-s + 0.432·5-s + (0.640 + 0.299i)6-s + (0.907 − 0.419i)7-s − 0.353i·8-s + (0.985 − 0.170i)9-s + (−0.264 − 0.152i)10-s − 1.68i·11-s + (−0.286 − 0.410i)12-s + (1.04 + 0.603i)13-s + (−0.704 − 0.0638i)14-s + (−0.431 + 0.0369i)15-s + (−0.125 + 0.216i)16-s + (−0.478 + 0.828i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.786 + 0.617i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.786 + 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.684791 - 0.236505i\)
\(L(\frac12)\) \(\approx\) \(0.684791 - 0.236505i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (1.72 - 0.148i)T \)
7 \( 1 + (-2.40 + 1.11i)T \)
good5 \( 1 - 0.967T + 5T^{2} \)
11 \( 1 + 5.57iT - 11T^{2} \)
13 \( 1 + (-3.76 - 2.17i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.97 - 3.41i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.86 + 2.23i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.65iT - 23T^{2} \)
29 \( 1 + (-4.61 + 2.66i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.34 - 3.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.243 - 0.421i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0818 - 0.141i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.35 + 7.53i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.74 - 8.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.74 - 1.00i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.836 - 1.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.47 + 2.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.72 - 4.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.64iT - 71T^{2} \)
73 \( 1 + (2.15 + 1.24i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.30 - 3.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.20 - 7.29i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.05 - 3.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.2 - 5.92i)T + (48.5 - 84.0i)T^{2} \)
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

−13.28781591092643270326612456291, −11.73192906544098792099424290151, −11.17410510686202181380956516815, −10.47764393270609479398146371832, −9.124143529977853348708408114793, −8.046571296751219428894374377022, −6.58814792565117609139707913370, −5.51015534050409771132588982704, −3.85361978627804289921231347667, −1.33487675771216378232646498621, 1.72797615657996853365772822925, 4.73372379870907654230378161101, 5.71720358037207070084742818901, 6.94968417373011233876566354725, 7.994502343094115015117972498042, 9.440372560053711601401422513741, 10.32827898205638285742477842583, 11.36753983428567392545033245285, 12.22588933737514349521865443585, 13.41820516328545710924196159740

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