↑

Properties

Label 2-132-44.7-c1-0-3
Degree $2$
Conductor $132$
Sign $-0.162 - 0.986i$
Analytic cond. $1.05402$
Root an. cond. $1.02665$
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  + (0.140 + 1.40i)2-s + (0.951 + 0.309i)3-s + (−1.96 + 0.394i)4-s + (−0.213 + 0.155i)5-s + (−0.301 + 1.38i)6-s + (1.03 + 3.17i)7-s + (−0.829 − 2.70i)8-s + (0.809 + 0.587i)9-s + (−0.248 − 0.278i)10-s + (−1.85 + 2.74i)11-s + (−1.98 − 0.231i)12-s + (3.89 − 5.36i)13-s + (−4.33 + 1.89i)14-s + (−0.251 + 0.0816i)15-s + (3.68 − 1.54i)16-s + (−2.74 − 3.78i)17-s + ⋯
L(s)  = 1  + (0.0990 + 0.995i)2-s + (0.549 + 0.178i)3-s + (−0.980 + 0.197i)4-s + (−0.0955 + 0.0694i)5-s + (−0.123 + 0.564i)6-s + (0.390 + 1.20i)7-s + (−0.293 − 0.956i)8-s + (0.269 + 0.195i)9-s + (−0.0785 − 0.0882i)10-s + (−0.560 + 0.828i)11-s + (−0.573 − 0.0666i)12-s + (1.08 − 1.48i)13-s + (−1.15 + 0.507i)14-s + (−0.0648 + 0.0210i)15-s + (0.922 − 0.386i)16-s + (−0.666 − 0.916i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 132 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.162 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(132\)    =    \(2^{2} \cdot 3 \cdot 11\)
Sign: $-0.162 - 0.986i$
Analytic conductor: \(1.05402\)
Root analytic conductor: \(1.02665\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{132} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 132,\ (\ :1/2),\ -0.162 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.787637 + 0.927513i\)
\(L(\frac12)\) \(\approx\) \(0.787637 + 0.927513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.140 - 1.40i)T \)
3 \( 1 + (-0.951 - 0.309i)T \)
11 \( 1 + (1.85 - 2.74i)T \)
good5 \( 1 + (0.213 - 0.155i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.03 - 3.17i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.89 + 5.36i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.74 + 3.78i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.27 + 3.92i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 0.765iT - 23T^{2} \)
29 \( 1 + (-3.67 + 1.19i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.37 + 3.27i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.274 - 0.843i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (5.66 + 1.83i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.10T + 43T^{2} \)
47 \( 1 + (4.91 + 1.59i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.59 - 3.33i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (9.46 - 3.07i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.87 - 2.57i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 5.99iT - 67T^{2} \)
71 \( 1 + (-2.69 - 3.71i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (14.3 - 4.65i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (11.6 + 8.47i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.496 + 0.360i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 2.78T + 89T^{2} \)
97 \( 1 + (-8.14 - 5.91i)T + (29.9 + 92.2i)T^{2} \)
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

−13.55908540288419332377882686566, −12.98001359105672290380181732764, −11.67002264771807896491927234161, −10.16510275900073777904074371113, −9.013067106873717049481437707350, −8.276243973288480677281751968445, −7.22950000716342658076811823763, −5.72556031586544322131711123165, −4.74413637678803812364222555510, −2.93454653218631189556218509170, 1.56706428127621530622917010929, 3.52724240215836083821119532710, 4.46566223983112134197322330629, 6.36936650098439884595880013503, 8.056655798799499949390772461218, 8.753652734319236187686331389395, 10.15631349483591574587284060815, 10.92416034388548460312009330681, 11.88492644454687275628527948203, 13.19274835987646695967187140304

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