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Properties

Label 2-168-56.27-c1-0-0
Degree $2$
Conductor $168$
Sign $-0.869 + 0.493i$
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  + (−0.185 + 1.40i)2-s − i·3-s + (−1.93 − 0.520i)4-s − 3.84·5-s + (1.40 + 0.185i)6-s + (−1.62 + 2.09i)7-s + (1.08 − 2.61i)8-s − 9-s + (0.713 − 5.38i)10-s − 4.54·11-s + (−0.520 + 1.93i)12-s + 1.81·13-s + (−2.63 − 2.66i)14-s + 3.84i·15-s + (3.45 + 2.00i)16-s + 3.49i·17-s + ⋯
L(s)  = 1  + (−0.131 + 0.991i)2-s − 0.577i·3-s + (−0.965 − 0.260i)4-s − 1.71·5-s + (0.572 + 0.0757i)6-s + (−0.612 + 0.790i)7-s + (0.384 − 0.923i)8-s − 0.333·9-s + (0.225 − 1.70i)10-s − 1.37·11-s + (−0.150 + 0.557i)12-s + 0.503·13-s + (−0.702 − 0.711i)14-s + 0.992i·15-s + (0.864 + 0.502i)16-s + 0.846i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 + 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.869 + 0.493i$
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: \((2,\ 168,\ (\ :1/2),\ -0.869 + 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0257874 - 0.0976357i\)
\(L(\frac12)\) \(\approx\) \(0.0257874 - 0.0976357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.185 - 1.40i)T \)
3 \( 1 + iT \)
7 \( 1 + (1.62 - 2.09i)T \)
good5 \( 1 + 3.84T + 5T^{2} \)
11 \( 1 + 4.54T + 11T^{2} \)
13 \( 1 - 1.81T + 13T^{2} \)
17 \( 1 - 3.49iT - 17T^{2} \)
19 \( 1 + 1.68iT - 19T^{2} \)
23 \( 1 + 5.00iT - 23T^{2} \)
29 \( 1 - 1.81iT - 29T^{2} \)
31 \( 1 + 5.34T + 31T^{2} \)
37 \( 1 - 1.42iT - 37T^{2} \)
41 \( 1 - 8.97iT - 41T^{2} \)
43 \( 1 + 8.03T + 43T^{2} \)
47 \( 1 + 4.83T + 47T^{2} \)
53 \( 1 + 5.87iT - 53T^{2} \)
59 \( 1 - 8.46iT - 59T^{2} \)
61 \( 1 + 3.01T + 61T^{2} \)
67 \( 1 + 4.42T + 67T^{2} \)
71 \( 1 + 1.47iT - 71T^{2} \)
73 \( 1 - 6.98iT - 73T^{2} \)
79 \( 1 - 2.97iT - 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 + 15.9iT - 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−13.07041496023963631787894708093, −12.74779317426382422451297583188, −11.53609492621542515281845624219, −10.34609520021505854337183603718, −8.700182990398512730828536666307, −8.189295089792743721717645877646, −7.24574253539907595475908731054, −6.16561534818555335967881264446, −4.82078530749890684424281534468, −3.30961397713345555387604849993, 0.096952769108891781697760662025, 3.20495338755381160887131786823, 3.93876544525517381672657956710, 5.13833608884063475774841035704, 7.40163621280048494032855683114, 8.141103272371197522874845914607, 9.386281923816895810023828589746, 10.49937289026895400304107329452, 11.10515910150750894844860967840, 12.01059727579875992327932603783

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