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Properties

Label 2-168-168.11-c1-0-12
Degree $2$
Conductor $168$
Sign $0.845 - 0.534i$
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 + (−1.64 − 0.531i)3-s + (1.53 + 1.27i)4-s + (0.646 + 1.11i)5-s + (−1.93 − 1.49i)6-s + (2.42 − 1.05i)7-s + (1.42 + 2.44i)8-s + (2.43 + 1.75i)9-s + (0.321 + 1.79i)10-s + (−1.60 − 0.923i)11-s + (−1.85 − 2.92i)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 + (−0.951 − 0.306i)3-s + (0.768 + 0.639i)4-s + (0.289 + 0.500i)5-s + (−0.790 − 0.612i)6-s + (0.917 − 0.397i)7-s + (0.505 + 0.862i)8-s + (0.811 + 0.584i)9-s + (0.101 + 0.569i)10-s + (−0.482 − 0.278i)11-s + (−0.535 − 0.844i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.845 - 0.534i$
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: \((2,\ 168,\ (\ :1/2),\ 0.845 - 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54595 + 0.447379i\)
\(L(\frac12)\) \(\approx\) \(1.54595 + 0.447379i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 0.531i)T \)
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

−13.07618740055425879517654147219, −11.80722575222157130277567115955, −11.16314010138596805917808265260, −10.44254673792066649788236324195, −8.468652593743422261780431251634, −7.14392600072558786702323362175, −6.55644939936829184795275901206, −5.22113958035220507376080916251, −4.37320314267718932638451097665, −2.26728656927119569268569916843, 1.81302812078921588774437685287, 4.00637852225618316028299508228, 5.17124577942917739978209212531, 5.73064601325779470066741349378, 7.13526651193785022305189098098, 8.708776277138180996873324289779, 10.25385045747044438719438336292, 10.79444741951174318735939169106, 11.88990093287339116559369330976, 12.57942403458252628064385124413

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