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Properties

Label 2-168-56.5-c2-0-22
Degree $2$
Conductor $168$
Sign $0.254 + 0.966i$
Analytic cond. $4.57766$
Root an. cond. $2.13954$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.96 − 0.353i)2-s + (0.866 + 1.5i)3-s + (3.74 + 1.39i)4-s + (3.07 − 5.32i)5-s + (−1.17 − 3.25i)6-s + (−5.56 − 4.25i)7-s + (−6.88 − 4.06i)8-s + (−1.5 + 2.59i)9-s + (−7.93 + 9.39i)10-s + (15.3 − 8.86i)11-s + (1.15 + 6.83i)12-s − 21.5·13-s + (9.44 + 10.3i)14-s + 10.6·15-s + (12.1 + 10.4i)16-s + (0.757 − 0.437i)17-s + ⋯
L(s)  = 1  + (−0.984 − 0.176i)2-s + (0.288 + 0.5i)3-s + (0.937 + 0.348i)4-s + (0.615 − 1.06i)5-s + (−0.195 − 0.543i)6-s + (−0.794 − 0.607i)7-s + (−0.861 − 0.508i)8-s + (−0.166 + 0.288i)9-s + (−0.793 + 0.939i)10-s + (1.39 − 0.805i)11-s + (0.0964 + 0.569i)12-s − 1.65·13-s + (0.674 + 0.738i)14-s + 0.710·15-s + (0.757 + 0.652i)16-s + (0.0445 − 0.0257i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.254 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.254 + 0.966i$
Analytic conductor: \(4.57766\)
Root analytic conductor: \(2.13954\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1),\ 0.254 + 0.966i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.789075 - 0.608047i\)
\(L(\frac12)\) \(\approx\) \(0.789075 - 0.608047i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 + 0.353i)T \)
3 \( 1 + (-0.866 - 1.5i)T \)
7 \( 1 + (5.56 + 4.25i)T \)
good5 \( 1 + (-3.07 + 5.32i)T + (-12.5 - 21.6i)T^{2} \)
11 \( 1 + (-15.3 + 8.86i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + 21.5T + 169T^{2} \)
17 \( 1 + (-0.757 + 0.437i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-12.0 + 20.8i)T + (-180.5 - 312. i)T^{2} \)
23 \( 1 + (-5.94 + 10.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 37.8iT - 841T^{2} \)
31 \( 1 + (-23.6 + 13.6i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (0.460 + 0.266i)T + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 23.0iT - 1.68e3T^{2} \)
43 \( 1 - 51.9iT - 1.84e3T^{2} \)
47 \( 1 + (-55.5 - 32.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-13.6 + 7.88i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (30.8 + 53.4i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (42.0 - 72.8i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-72.5 + 41.8i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 31.1T + 5.04e3T^{2} \)
73 \( 1 + (58.4 - 33.7i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (54.9 - 95.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 59.9T + 6.88e3T^{2} \)
89 \( 1 + (64.3 + 37.1i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 9.43iT - 9.40e3T^{2} \)
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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

−12.20269789045272482349569764965, −11.22321904261169060934500424691, −9.724545397014019249017405384757, −9.593008051839340592282885033125, −8.623782464811136539476135679661, −7.28902767131927341713000720873, −6.09386986931665868064175996760, −4.44336745220645738179573832651, −2.81056305769244599564626495854, −0.820373318729186821672325310980, 1.91329967902164301108558031716, 3.08141241960691402937199260021, 5.73443925906101667633636596491, 6.86770838584143638983006516927, 7.25246629330394058893776894427, 8.907127255561640019627246981610, 9.709534300950665958982885755544, 10.33814508355861275676973596638, 11.96024390190168544685026386389, 12.32189153847158001270437707278

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