Properties

Label 2-168-168.5-c1-0-2
Degree $2$
Conductor $168$
Sign $-0.952 - 0.305i$
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  + (0.174 + 1.40i)2-s + (0.618 + 1.61i)3-s + (−1.93 + 0.489i)4-s + (−2.46 − 1.42i)5-s + (−2.16 + 1.14i)6-s + (−1.02 + 2.44i)7-s + (−1.02 − 2.63i)8-s + (−2.23 + 2.00i)9-s + (1.57 − 3.71i)10-s + (2.42 + 4.20i)11-s + (−1.99 − 2.83i)12-s + 2.75·13-s + (−3.60 − 1.00i)14-s + (0.780 − 4.87i)15-s + (3.52 − 1.89i)16-s + (1.75 + 3.03i)17-s + ⋯
L(s)  = 1  + (0.123 + 0.992i)2-s + (0.356 + 0.934i)3-s + (−0.969 + 0.244i)4-s + (−1.10 − 0.637i)5-s + (−0.883 + 0.469i)6-s + (−0.385 + 0.922i)7-s + (−0.362 − 0.931i)8-s + (−0.745 + 0.666i)9-s + (0.496 − 1.17i)10-s + (0.731 + 1.26i)11-s + (−0.574 − 0.818i)12-s + 0.763·13-s + (−0.963 − 0.268i)14-s + (0.201 − 1.25i)15-s + (0.880 − 0.474i)16-s + (0.425 + 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.952 - 0.305i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148850 + 0.951457i\)
\(L(\frac12)\) \(\approx\) \(0.148850 + 0.951457i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.174 - 1.40i)T \)
3 \( 1 + (-0.618 - 1.61i)T \)
7 \( 1 + (1.02 - 2.44i)T \)
good5 \( 1 + (2.46 + 1.42i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.42 - 4.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.75T + 13T^{2} \)
17 \( 1 + (-1.75 - 3.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.14 + 5.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.15 - 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.90T + 29T^{2} \)
31 \( 1 + (0.858 - 0.495i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.06 - 0.614i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.10T + 41T^{2} \)
43 \( 1 - 5.11iT - 43T^{2} \)
47 \( 1 + (-5.61 + 9.72i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.00 - 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.890 + 0.514i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.24 + 2.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.02 + 2.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.75iT - 71T^{2} \)
73 \( 1 + (-0.291 + 0.168i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.80 - 4.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 0.138iT - 83T^{2} \)
89 \( 1 + (-0.580 + 1.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−13.31553701967501104817411362435, −12.36779620266598931176701661863, −11.41571871617959118607790724170, −9.736037450187798928979178102674, −9.012555758003547177491952657521, −8.277094138072261691484357728229, −7.07312742223068903209976252968, −5.53549813715434181180735773745, −4.50329988885137505341073685553, −3.51812770089477855226107331126, 0.932425899737755812913475970357, 3.25579334641627327916952058237, 3.74835735158957128701525852745, 5.96732781678518758617244252223, 7.28686579972778912206869648497, 8.207112539751032438850194403352, 9.313204537821797757425764042079, 10.75260461114004296440193362603, 11.43840532797571345960777379855, 12.18708756167874244863949877573

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