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Properties

Label 2-168-168.11-c1-0-21
Degree $2$
Conductor $168$
Sign $-0.0239 + 0.999i$
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  + (1.08 − 0.911i)2-s + (−1.31 + 1.12i)3-s + (0.338 − 1.97i)4-s + (−1.86 − 3.22i)5-s + (−0.397 + 2.41i)6-s + (2.46 − 0.962i)7-s + (−1.43 − 2.44i)8-s + (0.465 − 2.96i)9-s + (−4.94 − 1.78i)10-s + (2.26 + 1.31i)11-s + (1.77 + 2.97i)12-s + 3.57i·13-s + (1.78 − 3.28i)14-s + (6.07 + 2.14i)15-s + (−3.77 − 1.33i)16-s + (−0.186 − 0.107i)17-s + ⋯
L(s)  = 1  + (0.764 − 0.644i)2-s + (−0.760 + 0.649i)3-s + (0.169 − 0.985i)4-s + (−0.832 − 1.44i)5-s + (−0.162 + 0.986i)6-s + (0.931 − 0.363i)7-s + (−0.505 − 0.862i)8-s + (0.155 − 0.987i)9-s + (−1.56 − 0.565i)10-s + (0.684 + 0.395i)11-s + (0.511 + 0.859i)12-s + 0.990i·13-s + (0.477 − 0.878i)14-s + (1.56 + 0.554i)15-s + (−0.942 − 0.333i)16-s + (−0.0451 − 0.0260i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0239 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0239 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.0239 + 0.999i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.0239 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.889918 - 0.911515i\)
\(L(\frac12)\) \(\approx\) \(0.889918 - 0.911515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.08 + 0.911i)T \)
3 \( 1 + (1.31 - 1.12i)T \)
7 \( 1 + (-2.46 + 0.962i)T \)
good5 \( 1 + (1.86 + 3.22i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.26 - 1.31i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.57iT - 13T^{2} \)
17 \( 1 + (0.186 + 0.107i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.14 + 1.97i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.33 - 4.04i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.57T + 29T^{2} \)
31 \( 1 + (-4.26 - 2.46i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-7.31 + 4.22i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.909iT - 41T^{2} \)
43 \( 1 - 3.73T + 43T^{2} \)
47 \( 1 + (-0.586 - 1.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.06 + 1.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.79 - 3.92i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.301 - 0.173i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.98 - 8.63i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 + (-4.45 + 7.71i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.70 - 5.60i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.73iT - 83T^{2} \)
89 \( 1 + (15.4 - 8.93i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.88T + 97T^{2} \)
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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

−12.23683690667769115300951056189, −11.63093089309651038870706005266, −11.04333463687381138891811673068, −9.611582052891533654876981750311, −8.805121491496200666810731161592, −7.09319179199772663715574778145, −5.51784120446879526608702420651, −4.47355039245478056941110008759, −4.12116243297762397820393428232, −1.21558507659526598765097980844, 2.76832045584968138856812766369, 4.32641733510355934613041539716, 5.76260947889822686391021149597, 6.63153354721632569829210509869, 7.62730412362751043959614652393, 8.295179660138627361192860885194, 10.62314142673119155228371700388, 11.37977159024442697676187350655, 11.95270747470640014827516816232, 13.02790266408827329260805088543

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