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Properties

Label 2-168-168.5-c1-0-22
Degree $2$
Conductor $168$
Sign $0.495 + 0.868i$
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  + (1.30 − 0.550i)2-s + (−0.618 − 1.61i)3-s + (1.39 − 1.43i)4-s + (2.46 + 1.42i)5-s + (−1.69 − 1.76i)6-s + (−1.02 + 2.44i)7-s + (1.02 − 2.63i)8-s + (−2.23 + 2.00i)9-s + (4.00 + 0.497i)10-s + (−2.42 − 4.20i)11-s + (−3.18 − 1.36i)12-s − 2.75·13-s + (0.0151 + 3.74i)14-s + (0.780 − 4.87i)15-s + (−0.115 − 3.99i)16-s + (1.75 + 3.03i)17-s + ⋯
L(s)  = 1  + (0.921 − 0.389i)2-s + (−0.356 − 0.934i)3-s + (0.696 − 0.717i)4-s + (1.10 + 0.637i)5-s + (−0.692 − 0.721i)6-s + (−0.385 + 0.922i)7-s + (0.362 − 0.931i)8-s + (−0.745 + 0.666i)9-s + (1.26 + 0.157i)10-s + (−0.731 − 1.26i)11-s + (−0.918 − 0.394i)12-s − 0.763·13-s + (0.00403 + 0.999i)14-s + (0.201 − 1.25i)15-s + (−0.0289 − 0.999i)16-s + (0.425 + 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.495 + 0.868i$
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: \((2,\ 168,\ (\ :1/2),\ 0.495 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56085 - 0.906492i\)
\(L(\frac12)\) \(\approx\) \(1.56085 - 0.906492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.550i)T \)
3 \( 1 + (0.618 + 1.61i)T \)
7 \( 1 + (1.02 - 2.44i)T \)
good5 \( 1 + (-2.46 - 1.42i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (2.42 + 4.20i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.75T + 13T^{2} \)
17 \( 1 + (-1.75 - 3.03i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.14 - 5.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.15 - 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.90T + 29T^{2} \)
31 \( 1 + (0.858 - 0.495i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.06 + 0.614i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.10T + 41T^{2} \)
43 \( 1 + 5.11iT - 43T^{2} \)
47 \( 1 + (-5.61 + 9.72i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.00 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.890 - 0.514i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.24 - 2.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.02 - 2.90i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.75iT - 71T^{2} \)
73 \( 1 + (-0.291 + 0.168i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.80 - 4.85i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.138iT - 83T^{2} \)
89 \( 1 + (-0.580 + 1.00i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
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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

−12.68525819489786157084355760486, −11.93890205934456350909166738435, −10.76203722127171045066827911921, −10.06311036047444926015953799351, −8.435832713694190833520774661385, −6.90411658793784569598151997654, −5.85772082438099811732812306021, −5.54605547404688259267238303438, −3.05980165506910829198654803910, −2.04220936508282510792494909258, 2.73540641934814965628844489332, 4.61507390841455598001307839672, 4.97835062783547048067097657780, 6.35993807918435879365873370201, 7.44210323829325234421058122434, 9.154327536772111811281027051273, 10.02651610105918094791428622627, 10.89516534251911136263059904837, 12.29351745454970497540711840639, 13.01927105452087460574655384807

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