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Properties

Label 2-168-21.5-c1-0-5
Degree $2$
Conductor $168$
Sign $0.493 + 0.869i$
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  + (−1.67 + 0.441i)3-s + (1.40 − 2.43i)5-s + (−2.08 − 1.62i)7-s + (2.60 − 1.47i)9-s + (4.74 − 2.74i)11-s + 1.35i·13-s + (−1.27 + 4.69i)15-s + (−2.88 − 5.00i)17-s + (1.71 + 0.992i)19-s + (4.21 + 1.80i)21-s + (−2.09 − 1.21i)23-s + (−1.44 − 2.49i)25-s + (−3.71 + 3.63i)27-s + 7.05i·29-s + (−3.07 + 1.77i)31-s + ⋯
L(s)  = 1  + (−0.966 + 0.254i)3-s + (0.627 − 1.08i)5-s + (−0.788 − 0.615i)7-s + (0.869 − 0.493i)9-s + (1.43 − 0.826i)11-s + 0.376i·13-s + (−0.329 + 1.21i)15-s + (−0.700 − 1.21i)17-s + (0.394 + 0.227i)19-s + (0.919 + 0.393i)21-s + (−0.437 − 0.252i)23-s + (−0.288 − 0.499i)25-s + (−0.715 + 0.698i)27-s + 1.31i·29-s + (−0.552 + 0.318i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.493 + 0.869i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ 0.493 + 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.772504 - 0.450048i\)
\(L(\frac12)\) \(\approx\) \(0.772504 - 0.450048i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.67 - 0.441i)T \)
7 \( 1 + (2.08 + 1.62i)T \)
good5 \( 1 + (-1.40 + 2.43i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.74 + 2.74i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.35iT - 13T^{2} \)
17 \( 1 + (2.88 + 5.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.71 - 0.992i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.09 + 1.21i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.05iT - 29T^{2} \)
31 \( 1 + (3.07 - 1.77i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.14 - 3.71i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.81T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (-0.201 + 0.348i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.28 + 3.04i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.28 + 2.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.75 + 2.74i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.45 - 5.97i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.08iT - 71T^{2} \)
73 \( 1 + (0.295 - 0.170i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.19 + 2.06i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + (-0.576 + 0.998i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.0iT - 97T^{2} \)
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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

−12.53574466233292106455996805229, −11.71147936156489447828661915572, −10.67883892884511582079087487378, −9.432050796436887581358766099465, −9.040442449523770344223601996133, −7.03373781709739093808566228097, −6.16805290630852140360322635944, −5.02271833642966795620650288408, −3.82682366036790885042336687141, −1.04060118499167186243482119534, 2.15084230867860230027899069378, 4.03540025147548806552119052147, 5.89363648135415347138813107555, 6.38117604804195252677544804280, 7.36413984098407004932304082114, 9.221856710576984989159257336114, 10.06713589418775907777735006691, 10.97266565077193816614853574349, 12.00654234446597682394779334486, 12.76783219782184259959224189862

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