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Properties

Label 2-168-56.3-c1-0-14
Degree $2$
Conductor $168$
Sign $-0.788 + 0.614i$
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  + (0.582 − 1.28i)2-s + (−0.866 − 0.5i)3-s + (−1.32 − 1.50i)4-s + (−0.128 − 0.222i)5-s + (−1.14 + 0.824i)6-s + (−0.623 − 2.57i)7-s + (−2.70 + 0.828i)8-s + (0.499 + 0.866i)9-s + (−0.362 + 0.0360i)10-s + (1.79 − 3.10i)11-s + (0.393 + 1.96i)12-s − 4.57·13-s + (−3.67 − 0.693i)14-s + 0.257i·15-s + (−0.506 + 3.96i)16-s + (6.92 + 3.99i)17-s + ⋯
L(s)  = 1  + (0.411 − 0.911i)2-s + (−0.499 − 0.288i)3-s + (−0.660 − 0.750i)4-s + (−0.0575 − 0.0996i)5-s + (−0.468 + 0.336i)6-s + (−0.235 − 0.971i)7-s + (−0.956 + 0.293i)8-s + (0.166 + 0.288i)9-s + (−0.114 + 0.0113i)10-s + (0.540 − 0.936i)11-s + (0.113 + 0.566i)12-s − 1.27·13-s + (−0.982 − 0.185i)14-s + 0.0664i·15-s + (−0.126 + 0.991i)16-s + (1.68 + 0.970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 + 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.788 + 0.614i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.788 + 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.335692 - 0.976512i\)
\(L(\frac12)\) \(\approx\) \(0.335692 - 0.976512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.582 + 1.28i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (0.623 + 2.57i)T \)
good5 \( 1 + (0.128 + 0.222i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.79 + 3.10i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.57T + 13T^{2} \)
17 \( 1 + (-6.92 - 3.99i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.201 + 0.116i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.76 + 3.32i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.80iT - 29T^{2} \)
31 \( 1 + (1.03 - 1.79i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.46 + 3.73i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.55iT - 41T^{2} \)
43 \( 1 + 5.42T + 43T^{2} \)
47 \( 1 + (1.42 + 2.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.93 + 1.11i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.14 - 1.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.44 - 7.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.867 - 1.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.97iT - 71T^{2} \)
73 \( 1 + (6.57 + 3.79i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.51 - 4.34i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.79iT - 83T^{2} \)
89 \( 1 + (2.25 - 1.29i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.6iT - 97T^{2} \)
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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

−12.45995669163941227107329261655, −11.47340783353877212579919286513, −10.52975780352978355014753191722, −9.818472489451279851420432225839, −8.396644567987704569128838969871, −6.96380803546520227301414966384, −5.72549184855010252548775815091, −4.49426703486363547509261170030, −3.15833217541621921316448591893, −0.999129032479459601524553315695, 3.12073559716391403355490503007, 4.81979180489466703265764988463, 5.52646963464327259202793855549, 6.85945208729380876699444613851, 7.69700590053690780857950942327, 9.327602034338130167354021812889, 9.730099575101252565048046624576, 11.66684585051016685275143730887, 12.19827549681197529320972489529, 13.10031545944983860241548552297

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