Properties

Label 2-168-168.107-c1-0-17
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: $\chi_{168} (107, \cdot )$
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

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

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