Properties

Label 2-168-168.11-c1-0-9
Degree $2$
Conductor $168$
Sign $0.990 - 0.139i$
Analytic cond. $1.34148$
Root an. cond. $1.15822$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.32 − 0.481i)2-s + (1.28 + 1.16i)3-s + (1.53 + 1.27i)4-s + (−0.646 − 1.11i)5-s + (−1.14 − 2.16i)6-s + (2.42 − 1.05i)7-s + (−1.42 − 2.44i)8-s + (0.300 + 2.98i)9-s + (0.321 + 1.79i)10-s + (1.60 + 0.923i)11-s + (0.488 + 3.42i)12-s + 2.25i·13-s + (−3.73 + 0.232i)14-s + (0.470 − 2.18i)15-s + (0.726 + 3.93i)16-s + (3.89 + 2.24i)17-s + ⋯
L(s)  = 1  + (−0.940 − 0.340i)2-s + (0.741 + 0.670i)3-s + (0.768 + 0.639i)4-s + (−0.289 − 0.500i)5-s + (−0.469 − 0.883i)6-s + (0.917 − 0.397i)7-s + (−0.505 − 0.862i)8-s + (0.100 + 0.994i)9-s + (0.101 + 0.569i)10-s + (0.482 + 0.278i)11-s + (0.140 + 0.990i)12-s + 0.625i·13-s + (−0.998 + 0.0620i)14-s + (0.121 − 0.565i)15-s + (0.181 + 0.983i)16-s + (0.944 + 0.545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.990 - 0.139i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.990 - 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00786 + 0.0707345i\)
\(L(\frac12)\) \(\approx\) \(1.00786 + 0.0707345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.481i)T \)
3 \( 1 + (-1.28 - 1.16i)T \)
7 \( 1 + (-2.42 + 1.05i)T \)
good5 \( 1 + (0.646 + 1.11i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.60 - 0.923i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.25iT - 13T^{2} \)
17 \( 1 + (-3.89 - 2.24i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.80 + 4.86i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.519 - 0.900i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.32T + 29T^{2} \)
31 \( 1 + (3.69 + 2.13i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.18 - 4.72i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.39iT - 41T^{2} \)
43 \( 1 + 6.02T + 43T^{2} \)
47 \( 1 + (5.90 + 10.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.02 - 10.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.57 + 5.52i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.65 + 4.41i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.05 + 5.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.0T + 71T^{2} \)
73 \( 1 + (4.38 - 7.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.37 + 1.36i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.74iT - 83T^{2} \)
89 \( 1 + (8.31 - 4.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.73T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.61658597450042786518150940335, −11.53325106465660566470031693663, −10.69121849853065400162200843332, −9.693485835070928960320303071581, −8.723814692031372697727473753095, −8.092829498495045804097002617423, −6.95169394975746546994009500556, −4.80543825494522275007026849672, −3.64846271331490160132999193163, −1.82046617707922609398549009913, 1.62846660223588520052601682497, 3.23533215741948815079194790916, 5.55736671194473332537811383581, 6.83100103528438224628132193522, 7.80608591119707598715277656064, 8.432930752516202643876007932865, 9.490161762268278590744293091359, 10.70818174496861697700192141711, 11.67825014797114928091304712096, 12.59756890729206513269834659565

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