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Properties

Label 2-168-168.125-c3-0-60
Degree $2$
Conductor $168$
Sign $0.550 - 0.834i$
Analytic cond. $9.91232$
Root an. cond. $3.14838$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.65 + 0.964i)2-s + (5.10 − 0.953i)3-s + (6.13 + 5.13i)4-s + 17.0i·5-s + (14.5 + 2.39i)6-s + (−5.04 − 17.8i)7-s + (11.3 + 19.5i)8-s + (25.1 − 9.73i)9-s + (−16.4 + 45.2i)10-s + 37.8·11-s + (36.2 + 20.3i)12-s − 77.9·13-s + (3.76 − 52.2i)14-s + (16.2 + 86.8i)15-s + (11.3 + 62.9i)16-s − 34.6·17-s + ⋯
L(s)  = 1  + (0.940 + 0.341i)2-s + (0.983 − 0.183i)3-s + (0.767 + 0.641i)4-s + 1.52i·5-s + (0.986 + 0.162i)6-s + (−0.272 − 0.962i)7-s + (0.502 + 0.864i)8-s + (0.932 − 0.360i)9-s + (−0.518 + 1.42i)10-s + 1.03·11-s + (0.871 + 0.489i)12-s − 1.66·13-s + (0.0718 − 0.997i)14-s + (0.279 + 1.49i)15-s + (0.177 + 0.984i)16-s − 0.494·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.550 - 0.834i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ 0.550 - 0.834i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.50888 + 1.88797i\)
\(L(\frac12)\) \(\approx\) \(3.50888 + 1.88797i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.65 - 0.964i)T \)
3 \( 1 + (-5.10 + 0.953i)T \)
7 \( 1 + (5.04 + 17.8i)T \)
good5 \( 1 - 17.0iT - 125T^{2} \)
11 \( 1 - 37.8T + 1.33e3T^{2} \)
13 \( 1 + 77.9T + 2.19e3T^{2} \)
17 \( 1 + 34.6T + 4.91e3T^{2} \)
19 \( 1 - 37.7T + 6.85e3T^{2} \)
23 \( 1 + 47.8iT - 1.21e4T^{2} \)
29 \( 1 - 180.T + 2.43e4T^{2} \)
31 \( 1 + 163. iT - 2.97e4T^{2} \)
37 \( 1 + 159. iT - 5.06e4T^{2} \)
41 \( 1 - 81.5T + 6.89e4T^{2} \)
43 \( 1 + 241. iT - 7.95e4T^{2} \)
47 \( 1 + 356.T + 1.03e5T^{2} \)
53 \( 1 + 585.T + 1.48e5T^{2} \)
59 \( 1 - 172. iT - 2.05e5T^{2} \)
61 \( 1 - 572.T + 2.26e5T^{2} \)
67 \( 1 + 765. iT - 3.00e5T^{2} \)
71 \( 1 - 925. iT - 3.57e5T^{2} \)
73 \( 1 + 590. iT - 3.89e5T^{2} \)
79 \( 1 + 28.4T + 4.93e5T^{2} \)
83 \( 1 - 2.66iT - 5.71e5T^{2} \)
89 \( 1 + 600.T + 7.04e5T^{2} \)
97 \( 1 - 703. iT - 9.12e5T^{2} \)
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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

−12.73450252949969716173532271414, −11.67298441914560608060485693181, −10.52583468608226095325209514999, −9.582104403310800345488735515357, −7.85465952774330770469539872846, −7.02999193671350528089365813350, −6.55251355626476258236492271574, −4.42480947591222943771513338090, −3.37921963834979327948074708367, −2.35773630562939373611714627575, 1.55720657419888196469191383014, 2.93232812233527355654833451552, 4.46247405835277343327691458153, 5.16239859488318340570934183599, 6.75145038088083795685297460997, 8.224596249455444915510795833658, 9.318035387088198113682021707494, 9.820010591272459180531878866049, 11.69542891216207383578774505087, 12.40241494674776845923494483551

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