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Properties

Label 2-168-168.5-c1-0-19
Degree $2$
Conductor $168$
Sign $0.706 + 0.707i$
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  + (−0.707 + 1.22i)2-s + (1.5 − 0.866i)3-s + (−0.999 − 1.73i)4-s + (−3.62 − 2.09i)5-s + 2.44i·6-s + (1.62 − 2.09i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (5.12 − 2.95i)10-s + (0.0857 + 0.148i)11-s + (−2.99 − 1.73i)12-s + (1.41 + 3.46i)14-s − 7.24·15-s + (−2.00 + 3.46i)16-s + (2.12 + 3.67i)18-s + ⋯
L(s)  = 1  + (−0.499 + 0.866i)2-s + (0.866 − 0.499i)3-s + (−0.499 − 0.866i)4-s + (−1.61 − 0.935i)5-s + 0.999i·6-s + (0.612 − 0.790i)7-s + 0.999·8-s + (0.5 − 0.866i)9-s + (1.61 − 0.935i)10-s + (0.0258 + 0.0448i)11-s + (−0.866 − 0.500i)12-s + (0.377 + 0.925i)14-s − 1.87·15-s + (−0.500 + 0.866i)16-s + (0.499 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.706 + 0.707i$
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: \((2,\ 168,\ (\ :1/2),\ 0.706 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.834312 - 0.346049i\)
\(L(\frac12)\) \(\approx\) \(0.834312 - 0.346049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 1.22i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (-1.62 + 2.09i)T \)
good5 \( 1 + (3.62 + 2.09i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.0857 - 0.148i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 + (5.37 - 3.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.03 - 8.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.98 + 2.30i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-8.48 + 4.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.86 - 6.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.76iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.5iT - 97T^{2} \)
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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

−12.80849472640676188843890857589, −11.79681366978133850406297616697, −10.52625358378780646427886621938, −9.081614607006918424209978112388, −8.306887473294608186943884842203, −7.70343024205803142836580164284, −6.87418260018588715958806117078, −4.85394360141271860217538016938, −3.89308266053694064024751807343, −1.02144656163216996523133460833, 2.53490431193428624634432690307, 3.58602435242247003626243127541, 4.64103339255794857769084713322, 7.19625598872623055651045968226, 8.137653966365733706642113049803, 8.690658367883543879252176892806, 10.03970845039765290764869198924, 10.99289635652748039291487779932, 11.66319167707082652482350229044, 12.58519226030291841966644791953

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