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Properties

Label 2-168-56.27-c1-0-13
Degree $2$
Conductor $168$
Sign $0.544 + 0.839i$
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  + (1.33 − 0.474i)2-s − i·3-s + (1.54 − 1.26i)4-s − 1.58·5-s + (−0.474 − 1.33i)6-s + (2.37 + 1.15i)7-s + (1.46 − 2.42i)8-s − 9-s + (−2.10 + 0.750i)10-s − 2.26·11-s + (−1.26 − 1.54i)12-s − 0.548·13-s + (3.71 + 0.409i)14-s + 1.58i·15-s + (0.798 − 3.91i)16-s − 0.433i·17-s + ⋯
L(s)  = 1  + (0.941 − 0.335i)2-s − 0.577i·3-s + (0.774 − 0.632i)4-s − 0.706·5-s + (−0.193 − 0.543i)6-s + (0.899 + 0.436i)7-s + (0.517 − 0.855i)8-s − 0.333·9-s + (−0.665 + 0.237i)10-s − 0.681·11-s + (−0.365 − 0.447i)12-s − 0.152·13-s + (0.993 + 0.109i)14-s + 0.408i·15-s + (0.199 − 0.979i)16-s − 0.105i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.544 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.544 + 0.839i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.544 + 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58377 - 0.860556i\)
\(L(\frac12)\) \(\approx\) \(1.58377 - 0.860556i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.474i)T \)
3 \( 1 + iT \)
7 \( 1 + (-2.37 - 1.15i)T \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 + 2.26T + 11T^{2} \)
13 \( 1 + 0.548T + 13T^{2} \)
17 \( 1 + 0.433iT - 17T^{2} \)
19 \( 1 - 6.02iT - 19T^{2} \)
23 \( 1 - 8.24iT - 23T^{2} \)
29 \( 1 + 0.548iT - 29T^{2} \)
31 \( 1 - 7.50T + 31T^{2} \)
37 \( 1 + 4.21iT - 37T^{2} \)
41 \( 1 + 7.09iT - 41T^{2} \)
43 \( 1 + 1.82T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + 3.71iT - 53T^{2} \)
59 \( 1 + 11.5iT - 59T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 - 9.35T + 67T^{2} \)
71 \( 1 - 1.27iT - 71T^{2} \)
73 \( 1 + 0.867iT - 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 6.13iT - 83T^{2} \)
89 \( 1 - 7.95iT - 89T^{2} \)
97 \( 1 + 19.1iT - 97T^{2} \)
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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

−12.49794020178364222972757607283, −11.80312341490752834056689062626, −11.13816214408735581641470631249, −9.878185559972681959748298760115, −8.133752688044536737171495837295, −7.46866623244886268202343845171, −5.96396559338459231222415025737, −4.96337072797544825378143763595, −3.51599011731338244549462232509, −1.88743078755274910270504913449, 2.81904083217511134220079692237, 4.38286951338907608167278664126, 4.91618852077979261563307045377, 6.52312283240313957878862198686, 7.75852364273287657327217624652, 8.489854466145166326096668437929, 10.30072316593476333634056788856, 11.18872516303756367468196727344, 11.88950703753420643932188706046, 13.09104565212051897004592458199

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