Properties

Label 2-168-168.11-c1-0-3
Degree $2$
Conductor $168$
Sign $0.999 - 0.0114i$
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  + (−0.982 − 1.01i)2-s + (−0.987 − 1.42i)3-s + (−0.0680 + 1.99i)4-s + (1.77 + 3.07i)5-s + (−0.476 + 2.40i)6-s + (0.793 + 2.52i)7-s + (2.09 − 1.89i)8-s + (−1.04 + 2.81i)9-s + (1.38 − 4.83i)10-s + (−0.396 − 0.229i)11-s + (2.91 − 1.87i)12-s − 0.799i·13-s + (1.78 − 3.28i)14-s + (2.62 − 5.57i)15-s + (−3.99 − 0.271i)16-s + (5.48 + 3.16i)17-s + ⋯
L(s)  = 1  + (−0.694 − 0.719i)2-s + (−0.570 − 0.821i)3-s + (−0.0340 + 0.999i)4-s + (0.795 + 1.37i)5-s + (−0.194 + 0.980i)6-s + (0.299 + 0.953i)7-s + (0.742 − 0.670i)8-s + (−0.349 + 0.936i)9-s + (0.437 − 1.52i)10-s + (−0.119 − 0.0690i)11-s + (0.840 − 0.541i)12-s − 0.221i·13-s + (0.477 − 0.878i)14-s + (0.678 − 1.43i)15-s + (−0.997 − 0.0679i)16-s + (1.33 + 0.767i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.999 - 0.0114i$
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: \((2,\ 168,\ (\ :1/2),\ 0.999 - 0.0114i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.772137 + 0.00442539i\)
\(L(\frac12)\) \(\approx\) \(0.772137 + 0.00442539i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.982 + 1.01i)T \)
3 \( 1 + (0.987 + 1.42i)T \)
7 \( 1 + (-0.793 - 2.52i)T \)
good5 \( 1 + (-1.77 - 3.07i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.396 + 0.229i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.799iT - 13T^{2} \)
17 \( 1 + (-5.48 - 3.16i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.61 + 4.53i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.55 - 2.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.60T + 29T^{2} \)
31 \( 1 + (-4.42 - 2.55i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.89 - 1.67i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 - 3.56T + 43T^{2} \)
47 \( 1 + (2.15 + 3.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.16 + 2.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.49 + 0.864i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.60 + 2.66i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.00979 + 0.0169i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.04T + 71T^{2} \)
73 \( 1 + (-4.15 + 7.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.8 - 6.28i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.694iT - 83T^{2} \)
89 \( 1 + (-7.02 + 4.05i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.05T + 97T^{2} \)
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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

−12.56353618694170397673143502528, −11.65764502781934823287100843949, −10.81594317554563917924559636465, −10.10652364406191742007846537214, −8.760219697270913078864284926170, −7.60548967799814840400790568304, −6.59873794018834260508015337385, −5.48284928235432705841155200946, −3.00930964797720064163324419513, −1.92709563877778171617926670832, 1.09058290625794011426296884400, 4.40296521138567192703956817661, 5.26397591698371328829243171134, 6.23339438475832082105932705333, 7.76758963970938187009544135030, 8.856909037709107639249469611993, 9.809134476371044028786496383516, 10.31211288107932690321670809839, 11.59194104243450857512564265189, 12.84470007806998910987663742892

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