Properties

Label 2-168-168.101-c1-0-13
Degree $2$
Conductor $168$
Sign $0.998 + 0.0532i$
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  + (−1.41 + 0.0117i)2-s + (1.70 + 0.298i)3-s + (1.99 − 0.0332i)4-s + (0.337 − 0.195i)5-s + (−2.41 − 0.401i)6-s + (1.39 − 2.24i)7-s + (−2.82 + 0.0704i)8-s + (2.82 + 1.01i)9-s + (−0.475 + 0.279i)10-s + (−0.748 + 1.29i)11-s + (3.42 + 0.539i)12-s − 3.28·13-s + (−1.94 + 3.19i)14-s + (0.634 − 0.232i)15-s + (3.99 − 0.132i)16-s + (1.68 − 2.91i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.00830i)2-s + (0.985 + 0.172i)3-s + (0.999 − 0.0166i)4-s + (0.151 − 0.0872i)5-s + (−0.986 − 0.164i)6-s + (0.527 − 0.849i)7-s + (−0.999 + 0.0249i)8-s + (0.940 + 0.339i)9-s + (−0.150 + 0.0884i)10-s + (−0.225 + 0.390i)11-s + (0.987 + 0.155i)12-s − 0.911·13-s + (−0.520 + 0.853i)14-s + (0.163 − 0.0599i)15-s + (0.999 − 0.0332i)16-s + (0.407 − 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.998 + 0.0532i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.998 + 0.0532i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07773 - 0.0287300i\)
\(L(\frac12)\) \(\approx\) \(1.07773 - 0.0287300i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0117i)T \)
3 \( 1 + (-1.70 - 0.298i)T \)
7 \( 1 + (-1.39 + 2.24i)T \)
good5 \( 1 + (-0.337 + 0.195i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.748 - 1.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
17 \( 1 + (-1.68 + 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.56 - 4.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.72 + 2.72i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.13T + 29T^{2} \)
31 \( 1 + (3.60 + 2.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.46 - 4.31i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.1T + 41T^{2} \)
43 \( 1 + 4.79iT - 43T^{2} \)
47 \( 1 + (-2.51 - 4.34i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.499 - 0.864i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.36 - 0.785i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.40 + 5.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.05 + 1.76i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.3iT - 71T^{2} \)
73 \( 1 + (-2.76 - 1.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.239 + 0.414i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 17.4iT - 83T^{2} \)
89 \( 1 + (2.54 + 4.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.00iT - 97T^{2} \)
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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

−12.76310683055891619734531514213, −11.59614606753694214450449962236, −10.34089225588404086330865301359, −9.788101050462972437161899094040, −8.752766291162401213947438615329, −7.58467636887821767979787058661, −7.18258368315078546569180846375, −5.10335056432973245141278142344, −3.36634340466016723789434989559, −1.75903261884789458372009146207, 1.92643701495801372284142810891, 3.14808553724927076822090342786, 5.36022266642475431296745702089, 6.92123407146629599660262085647, 7.85844270857429233932795764971, 8.778348099808517027107949063350, 9.489254138092792408271890476990, 10.57748919066392090166762848974, 11.75364154098709137659619536270, 12.64752905990941372931315448083

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