Properties

Label 2-168-168.125-c1-0-9
Degree $2$
Conductor $168$
Sign $-0.239 - 0.970i$
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  + (1.22 + 0.707i)2-s + (−1 + 1.41i)3-s + (0.999 + 1.73i)4-s + 1.41i·5-s + (−2.22 + 1.02i)6-s + (−2 − 1.73i)7-s + 2.82i·8-s + (−1.00 − 2.82i)9-s + (−1.00 + 1.73i)10-s + 2.44·11-s + (−3.44 − 0.317i)12-s + 2·13-s + (−1.22 − 3.53i)14-s + (−2.00 − 1.41i)15-s + (−2.00 + 3.46i)16-s + 7.34·17-s + ⋯
L(s)  = 1  + (0.866 + 0.499i)2-s + (−0.577 + 0.816i)3-s + (0.499 + 0.866i)4-s + 0.632i·5-s + (−0.908 + 0.418i)6-s + (−0.755 − 0.654i)7-s + 0.999i·8-s + (−0.333 − 0.942i)9-s + (−0.316 + 0.547i)10-s + 0.738·11-s + (−0.995 − 0.0917i)12-s + 0.554·13-s + (−0.327 − 0.944i)14-s + (−0.516 − 0.365i)15-s + (−0.500 + 0.866i)16-s + 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.239 - 0.970i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.239 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.916289 + 1.16947i\)
\(L(\frac12)\) \(\approx\) \(0.916289 + 1.16947i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (1 - 1.41i)T \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 - 1.41iT - 5T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 7.34T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 + 4.89T + 29T^{2} \)
31 \( 1 + 6.92iT - 31T^{2} \)
37 \( 1 + 10.3iT - 37T^{2} \)
41 \( 1 + 2.44T + 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + 4.89T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 5.65iT - 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 3.46iT - 67T^{2} \)
71 \( 1 - 1.41iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 11.3iT - 83T^{2} \)
89 \( 1 + 7.34T + 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−13.10503247988220119478088356009, −12.13986436165695299118877226767, −11.14335581161909853164834082872, −10.32156628455590065379933269765, −9.132559562035687071910265120089, −7.51606245150690005120117675849, −6.45119263188054980877002830287, −5.66910146181115273914855359390, −4.07787527264030149165496335425, −3.35602937821452338064137109669, 1.41536962951545678532655654874, 3.24273203086242754636273006788, 4.95317988287184884838441410231, 5.98020452026363089909845191563, 6.76400738360322310421697647590, 8.391364747251461068586532976648, 9.682761874461389458329066923952, 10.84721656613308064826368959127, 11.98344484433092942705410143962, 12.43483383671597707684754132440

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