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Properties

Label 2-168-7.4-c3-0-8
Degree $2$
Conductor $168$
Sign $0.0648 + 0.997i$
Analytic cond. $9.91232$
Root an. cond. $3.14838$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.5 − 2.59i)3-s + (3.36 + 5.83i)5-s + (−11.2 − 14.7i)7-s + (−4.5 − 7.79i)9-s + (26.8 − 46.4i)11-s + 1.73·13-s + 20.2·15-s + (23.4 − 40.6i)17-s + (10.0 + 17.4i)19-s + (−55.1 + 7.10i)21-s + (−59.4 − 103. i)23-s + (39.8 − 68.9i)25-s − 27·27-s − 103.·29-s + (78.7 − 136. i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.301 + 0.521i)5-s + (−0.606 − 0.794i)7-s + (−0.166 − 0.288i)9-s + (0.735 − 1.27i)11-s + 0.0370·13-s + 0.347·15-s + (0.334 − 0.580i)17-s + (0.121 + 0.210i)19-s + (−0.572 + 0.0738i)21-s + (−0.539 − 0.933i)23-s + (0.318 − 0.551i)25-s − 0.192·27-s − 0.663·29-s + (0.456 − 0.790i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0648 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0648 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.0648 + 0.997i$
Analytic conductor: \(9.91232\)
Root analytic conductor: \(3.14838\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :3/2),\ 0.0648 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.24434 - 1.16607i\)
\(L(\frac12)\) \(\approx\) \(1.24434 - 1.16607i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 + 2.59i)T \)
7 \( 1 + (11.2 + 14.7i)T \)
good5 \( 1 + (-3.36 - 5.83i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-26.8 + 46.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 1.73T + 2.19e3T^{2} \)
17 \( 1 + (-23.4 + 40.6i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-10.0 - 17.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (59.4 + 103. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 103.T + 2.43e4T^{2} \)
31 \( 1 + (-78.7 + 136. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-18.8 - 32.6i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 287.T + 6.89e4T^{2} \)
43 \( 1 - 504.T + 7.95e4T^{2} \)
47 \( 1 + (-110. - 190. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (146. - 253. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-297. + 516. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-132. - 229. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (468. - 811. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 545.T + 3.57e5T^{2} \)
73 \( 1 + (149. - 259. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-470. - 814. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 611.T + 5.71e5T^{2} \)
89 \( 1 + (-588. - 1.01e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.48e3T + 9.12e5T^{2} \)
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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

−12.13682268978806284258307342076, −11.07433462919022285521698025737, −10.10189182606494935897088993841, −9.033979743666679156369892274538, −7.85305991945285947568860687789, −6.72908117699135860307815230480, −5.96447954199102087643182302365, −3.96052603486089578250394012369, −2.76588164870414093192336457608, −0.791082907314467231340483911723, 1.86037056881270356515241278859, 3.53229044572082582385760570616, 4.89247121082086081964756986731, 6.02731707305570619000326008367, 7.39263927165586063750492912785, 8.840384383041143502844814117044, 9.425597204679149622267515815888, 10.29497065193291410506644897672, 11.77374112606812147035844927900, 12.52710517653473720126366305216

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