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Properties

Label 2-168-56.27-c1-0-11
Degree $2$
Conductor $168$
Sign $-0.261 + 0.965i$
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  + (−0.185 − 1.40i)2-s − i·3-s + (−1.93 + 0.520i)4-s + 3.84·5-s + (−1.40 + 0.185i)6-s + (1.62 − 2.09i)7-s + (1.08 + 2.61i)8-s − 9-s + (−0.713 − 5.38i)10-s − 4.54·11-s + (0.520 + 1.93i)12-s − 1.81·13-s + (−3.23 − 1.88i)14-s − 3.84i·15-s + (3.45 − 2.00i)16-s + 3.49i·17-s + ⋯
L(s)  = 1  + (−0.131 − 0.991i)2-s − 0.577i·3-s + (−0.965 + 0.260i)4-s + 1.71·5-s + (−0.572 + 0.0757i)6-s + (0.612 − 0.790i)7-s + (0.384 + 0.923i)8-s − 0.333·9-s + (−0.225 − 1.70i)10-s − 1.37·11-s + (0.150 + 0.557i)12-s − 0.503·13-s + (−0.863 − 0.503i)14-s − 0.992i·15-s + (0.864 − 0.502i)16-s + 0.846i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.261 + 0.965i$
Analytic conductor: \(1.34148\)
Root analytic conductor: \(1.15822\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :1/2),\ -0.261 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.742018 - 0.970153i\)
\(L(\frac12)\) \(\approx\) \(0.742018 - 0.970153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.185 + 1.40i)T \)
3 \( 1 + iT \)
7 \( 1 + (-1.62 + 2.09i)T \)
good5 \( 1 - 3.84T + 5T^{2} \)
11 \( 1 + 4.54T + 11T^{2} \)
13 \( 1 + 1.81T + 13T^{2} \)
17 \( 1 - 3.49iT - 17T^{2} \)
19 \( 1 + 1.68iT - 19T^{2} \)
23 \( 1 - 5.00iT - 23T^{2} \)
29 \( 1 + 1.81iT - 29T^{2} \)
31 \( 1 - 5.34T + 31T^{2} \)
37 \( 1 + 1.42iT - 37T^{2} \)
41 \( 1 - 8.97iT - 41T^{2} \)
43 \( 1 + 8.03T + 43T^{2} \)
47 \( 1 - 4.83T + 47T^{2} \)
53 \( 1 - 5.87iT - 53T^{2} \)
59 \( 1 - 8.46iT - 59T^{2} \)
61 \( 1 - 3.01T + 61T^{2} \)
67 \( 1 + 4.42T + 67T^{2} \)
71 \( 1 - 1.47iT - 71T^{2} \)
73 \( 1 - 6.98iT - 73T^{2} \)
79 \( 1 + 2.97iT - 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 + 15.9iT - 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−12.73919832982095361591331972726, −11.39984614782715619070196050757, −10.36541113138842374469363225392, −9.882836668975268401790708905522, −8.547430955099264982981820142004, −7.46912791735707080031109524565, −5.83351059594051212677776957859, −4.79669305692102020462215924952, −2.71571812567618162852117446243, −1.54742883287721504472994871636, 2.46491380305963473045524775153, 5.01255061582048498819200376616, 5.36550930476513039678646565963, 6.55164190988669196319392380657, 8.072320240787188643399665894658, 9.035108328039973717966425613466, 9.910713590713552817499522555530, 10.59480811126995830307583115016, 12.36659257008600182369742670165, 13.44422855751985934954281314389

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