Properties

Degree $2$
Conductor $168$
Sign $0.0158 + 0.999i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−1.63 − 2.83i)5-s + (−1.5 − 2.17i)7-s + (−0.499 − 0.866i)9-s + (−1.63 + 2.83i)11-s + 6.27·13-s − 3.27·15-s + (2 − 3.46i)17-s + (3.13 + 5.43i)19-s + (−2.63 + 0.209i)21-s + (−2 − 3.46i)23-s + (−2.86 + 4.95i)25-s − 0.999·27-s + 5.27·29-s + (0.5 − 0.866i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.732 − 1.26i)5-s + (−0.566 − 0.823i)7-s + (−0.166 − 0.288i)9-s + (−0.493 + 0.855i)11-s + 1.74·13-s − 0.845·15-s + (0.485 − 0.840i)17-s + (0.719 + 1.24i)19-s + (−0.575 + 0.0456i)21-s + (−0.417 − 0.722i)23-s + (−0.572 + 0.991i)25-s − 0.192·27-s + 0.979·29-s + (0.0898 − 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0158 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0158 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.0158 + 0.999i$
Motivic weight: \(1\)
Character: $\chi_{168} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.0158 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755810 - 0.743946i\)
\(L(\frac12)\) \(\approx\) \(0.755810 - 0.743946i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (1.5 + 2.17i)T \)
good5 \( 1 + (1.63 + 2.83i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.63 - 2.83i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.27T + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.13 - 5.43i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.27T + 29T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.13 - 1.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.54T + 41T^{2} \)
43 \( 1 - 0.274T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.63 - 8.03i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.637 + 1.10i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.137 + 0.238i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (-2.13 + 3.70i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.77 + 10.0i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.27T + 83T^{2} \)
89 \( 1 + (-5.27 - 9.13i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.72T + 97T^{2} \)
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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

−12.55101789207877130792679053503, −11.90945748248805802556409302081, −10.50739952010498938262130732539, −9.394922777227781402507320912305, −8.248039147669915871251344272830, −7.59830591293046006325353882365, −6.21992685391274186500494114986, −4.66836700088757003073783697242, −3.49941947805212402969333979308, −1.09291297764788396020458372580, 2.97240022167920627026528298975, 3.66089444704539507203586502099, 5.63810465534148831552591221915, 6.65159635195609449237979103670, 8.049835981511005071143283555717, 8.876443902352536423344811958797, 10.21090441376872129078725427074, 11.04755870729834060697208629278, 11.74239549734248389398112833984, 13.22303832855252912954438821210

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