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Properties

Label 2-168-24.11-c1-0-4
Degree $2$
Conductor $168$
Sign $0.408 - 0.912i$
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 + i)2-s + (−1.61 + 0.618i)3-s − 2i·4-s + 3.23·5-s + (1 − 2.23i)6-s − i·7-s + (2 + 2i)8-s + (2.23 − 2.00i)9-s + (−3.23 + 3.23i)10-s + 4.47i·11-s + (1.23 + 3.23i)12-s + 1.23i·13-s + (1 + i)14-s + (−5.23 + 2.00i)15-s − 4·16-s + 2i·17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.934 + 0.356i)3-s − i·4-s + 1.44·5-s + (0.408 − 0.912i)6-s − 0.377i·7-s + (0.707 + 0.707i)8-s + (0.745 − 0.666i)9-s + (−1.02 + 1.02i)10-s + 1.34i·11-s + (0.356 + 0.934i)12-s + 0.342i·13-s + (0.267 + 0.267i)14-s + (−1.35 + 0.516i)15-s − 16-s + 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.408 - 0.912i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.660398 + 0.428091i\)
\(L(\frac12)\) \(\approx\) \(0.660398 + 0.428091i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (1.61 - 0.618i)T \)
7 \( 1 + iT \)
good5 \( 1 - 3.23T + 5T^{2} \)
11 \( 1 - 4.47iT - 11T^{2} \)
13 \( 1 - 1.23iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 - 7.23T + 19T^{2} \)
23 \( 1 + 0.472T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8.94iT - 31T^{2} \)
37 \( 1 - 6.47iT - 37T^{2} \)
41 \( 1 + 4.47iT - 41T^{2} \)
43 \( 1 + 0.472T + 43T^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + 1.23iT - 59T^{2} \)
61 \( 1 + 11.7iT - 61T^{2} \)
67 \( 1 + 10.9T + 67T^{2} \)
71 \( 1 + 6.94T + 71T^{2} \)
73 \( 1 + 0.472T + 73T^{2} \)
79 \( 1 - 4.94iT - 79T^{2} \)
83 \( 1 + 13.2iT - 83T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 - 3.52T + 97T^{2} \)
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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

−13.11643446133756396997049296737, −11.78108267893950617395837789255, −10.58021962214122083538488231234, −9.751604440311211858032999943428, −9.433652870650190990255859393548, −7.55231197147093563116712731035, −6.55474656977322621868955558402, −5.65933497744316834968327672220, −4.65734444428127727232820260793, −1.64045002354922689953542363738, 1.28560975224501459956908239469, 2.93610656398990764846680304930, 5.21780222039974953842592580844, 6.12883514255820122851315203702, 7.42816283885344141776152917366, 8.817553996952827156127671728618, 9.766310917196345536875931961327, 10.63076672700118197791280794840, 11.49742632682980316533362920978, 12.41677283199344727065792238022

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