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Properties

Label 2-168-24.11-c1-0-10
Degree $2$
Conductor $168$
Sign $-0.361 + 0.932i$
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  + (−1.34 − 0.430i)2-s + (−1.46 + 0.916i)3-s + (1.62 + 1.15i)4-s − 1.83·5-s + (2.37 − 0.602i)6-s − i·7-s + (−1.69 − 2.26i)8-s + (1.31 − 2.69i)9-s + (2.46 + 0.789i)10-s + (−3.45 − 0.210i)12-s − 5.57i·13-s + (−0.430 + 1.34i)14-s + (2.69 − 1.68i)15-s + (1.30 + 3.77i)16-s − 6.08i·17-s + (−2.93 + 3.06i)18-s + ⋯
L(s)  = 1  + (−0.952 − 0.304i)2-s + (−0.848 + 0.529i)3-s + (0.814 + 0.579i)4-s − 0.819·5-s + (0.969 − 0.245i)6-s − 0.377i·7-s + (−0.599 − 0.800i)8-s + (0.439 − 0.898i)9-s + (0.780 + 0.249i)10-s + (−0.998 − 0.0608i)12-s − 1.54i·13-s + (−0.115 + 0.360i)14-s + (0.695 − 0.433i)15-s + (0.327 + 0.944i)16-s − 1.47i·17-s + (−0.692 + 0.721i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.361 + 0.932i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.361 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175966 - 0.257046i\)
\(L(\frac12)\) \(\approx\) \(0.175966 - 0.257046i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.34 + 0.430i)T \)
3 \( 1 + (1.46 - 0.916i)T \)
7 \( 1 + iT \)
good5 \( 1 + 1.83T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 5.57iT - 13T^{2} \)
17 \( 1 + 6.08iT - 17T^{2} \)
19 \( 1 + 6.93T + 19T^{2} \)
23 \( 1 - 0.697T + 23T^{2} \)
29 \( 1 - 7.11T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 1.87iT - 37T^{2} \)
41 \( 1 - 4.69iT - 41T^{2} \)
43 \( 1 - 1.36T + 43T^{2} \)
47 \( 1 + 3.44T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + 8.94iT - 59T^{2} \)
61 \( 1 - 4.30iT - 61T^{2} \)
67 \( 1 - 4.51T + 67T^{2} \)
71 \( 1 + 6.08T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 11.7iT - 79T^{2} \)
83 \( 1 + 0.438iT - 83T^{2} \)
89 \( 1 - 2.64iT - 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−12.14731801865954211924935322307, −11.26112144227860717936554671519, −10.55411926497439898368575456318, −9.710905336671767982709332966196, −8.409437757790408034118481762401, −7.42258219398058971951440628215, −6.26911247990740458963250458215, −4.63025013210786777341270632099, −3.20692077994085836371351287468, −0.42892201712046555326719289753, 1.88046567193304654987738991144, 4.43152898310474941800042136230, 6.10808917657276020033904666109, 6.77537249003668918442478652313, 7.992875918426305277489913942361, 8.786579797698587278844111419017, 10.24942362526500631062315558356, 11.13099111090405994244682963337, 11.88144525432949124963360231946, 12.69887492103049170493243577807

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