Properties

Label 2-168-168.101-c1-0-11
Degree $2$
Conductor $168$
Sign $0.954 - 0.297i$
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  + (−1.07 + 0.917i)2-s + (1.52 − 0.815i)3-s + (0.317 − 1.97i)4-s + (−0.461 + 0.266i)5-s + (−0.896 + 2.27i)6-s + (0.489 + 2.60i)7-s + (1.46 + 2.41i)8-s + (1.66 − 2.49i)9-s + (0.252 − 0.710i)10-s + (2.28 − 3.96i)11-s + (−1.12 − 3.27i)12-s + 4.97·13-s + (−2.91 − 2.35i)14-s + (−0.487 + 0.783i)15-s + (−3.79 − 1.25i)16-s + (−2.16 + 3.74i)17-s + ⋯
L(s)  = 1  + (−0.761 + 0.648i)2-s + (0.882 − 0.470i)3-s + (0.158 − 0.987i)4-s + (−0.206 + 0.119i)5-s + (−0.365 + 0.930i)6-s + (0.184 + 0.982i)7-s + (0.519 + 0.854i)8-s + (0.556 − 0.830i)9-s + (0.0798 − 0.224i)10-s + (0.689 − 1.19i)11-s + (−0.325 − 0.945i)12-s + 1.38·13-s + (−0.778 − 0.628i)14-s + (−0.125 + 0.202i)15-s + (−0.949 − 0.313i)16-s + (−0.524 + 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.954 - 0.297i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.954 - 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06962 + 0.162977i\)
\(L(\frac12)\) \(\approx\) \(1.06962 + 0.162977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 - 0.917i)T \)
3 \( 1 + (-1.52 + 0.815i)T \)
7 \( 1 + (-0.489 - 2.60i)T \)
good5 \( 1 + (0.461 - 0.266i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.28 + 3.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.97T + 13T^{2} \)
17 \( 1 + (2.16 - 3.74i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.921 + 1.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.103 + 0.0596i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.74T + 29T^{2} \)
31 \( 1 + (1.93 + 1.11i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.02 - 4.05i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 + 1.87iT - 43T^{2} \)
47 \( 1 + (2.91 + 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.29 - 3.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.71 - 3.29i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.07 + 1.86i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.4 + 6.05i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.20iT - 71T^{2} \)
73 \( 1 + (-8.35 - 4.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.0228 + 0.0396i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.86iT - 83T^{2} \)
89 \( 1 + (-8.23 - 14.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 7.18iT - 97T^{2} \)
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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

−13.14865583435983778560423246456, −11.64269169002837303018980024159, −10.83133267051344804402986485803, −9.181774203849325090453127200899, −8.758612359397187435878579373918, −7.955121378593891296540204706616, −6.59542682924879892492849784921, −5.77946712437420161772796520285, −3.60991428368120974379294050432, −1.72004059241970968593884806242, 1.78999545394517107529480859481, 3.63345800120135284107643137387, 4.38866299573032592976048591472, 6.94533109781037198276468629735, 7.81301736583093023221261490547, 8.869251235159709722128828645704, 9.674722637632235744724563982106, 10.61049964154432719570484761456, 11.45875263727192773105897763930, 12.76521499314390767931886369689

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