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Properties

Label 2-168-7.3-c4-0-14
Degree $2$
Conductor $168$
Sign $-0.768 + 0.639i$
Analytic cond. $17.3661$
Root an. cond. $4.16727$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 − 2.59i)3-s + (23.7 − 13.7i)5-s + (−26.3 − 41.3i)7-s + (13.5 + 23.3i)9-s + (68.5 − 118. i)11-s + 248. i·13-s − 142.·15-s + (−376. − 217. i)17-s + (302. − 174. i)19-s + (11.2 + 254. i)21-s + (−57.6 − 99.8i)23-s + (64.8 − 112. i)25-s − 140. i·27-s − 131.·29-s + (−1.46e3 − 844. i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (0.951 − 0.549i)5-s + (−0.537 − 0.843i)7-s + (0.166 + 0.288i)9-s + (0.566 − 0.981i)11-s + 1.47i·13-s − 0.634·15-s + (−1.30 − 0.752i)17-s + (0.838 − 0.483i)19-s + (0.0254 + 0.576i)21-s + (−0.108 − 0.188i)23-s + (0.103 − 0.179i)25-s − 0.192i·27-s − 0.155·29-s + (−1.52 − 0.879i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $-0.768 + 0.639i$
Analytic conductor: \(17.3661\)
Root analytic conductor: \(4.16727\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 168,\ (\ :2),\ -0.768 + 0.639i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.177348568\)
\(L(\frac12)\) \(\approx\) \(1.177348568\)
\(L(3)\) 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 \( 1 + (4.5 + 2.59i)T \)
7 \( 1 + (26.3 + 41.3i)T \)
good5 \( 1 + (-23.7 + 13.7i)T + (312.5 - 541. i)T^{2} \)
11 \( 1 + (-68.5 + 118. i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 - 248. iT - 2.85e4T^{2} \)
17 \( 1 + (376. + 217. i)T + (4.17e4 + 7.23e4i)T^{2} \)
19 \( 1 + (-302. + 174. i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (57.6 + 99.8i)T + (-1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + 131.T + 7.07e5T^{2} \)
31 \( 1 + (1.46e3 + 844. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (1.06e3 + 1.85e3i)T + (-9.37e5 + 1.62e6i)T^{2} \)
41 \( 1 + 1.14e3iT - 2.82e6T^{2} \)
43 \( 1 + 2.64e3T + 3.41e6T^{2} \)
47 \( 1 + (-1.32e3 + 765. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (1.83e3 - 3.17e3i)T + (-3.94e6 - 6.83e6i)T^{2} \)
59 \( 1 + (-1.01e3 - 587. i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (-4.60e3 + 2.66e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-2.93e3 + 5.08e3i)T + (-1.00e7 - 1.74e7i)T^{2} \)
71 \( 1 - 177.T + 2.54e7T^{2} \)
73 \( 1 + (-698. - 403. i)T + (1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (443. + 767. i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 - 9.77e3iT - 4.74e7T^{2} \)
89 \( 1 + (-2.51e3 + 1.45e3i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 6.12e3iT - 8.85e7T^{2} \)
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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

−11.60141533608544728834894472458, −10.90058245438694822712623638897, −9.472094306301634732946889220494, −9.016939190699614323786563518150, −7.18625081496426750198154596934, −6.43439294700855468957324474108, −5.27010466929643940597994689253, −3.91109454746424779790825347982, −1.90549496914284730039998311770, −0.45479499169504815764238808307, 1.88488061151090743019496087524, 3.35283596083093624671581783980, 5.13278880767390391513299311077, 6.05404681224248130187258893429, 6.94805665274422607172885623392, 8.596268934248350312985808057057, 9.790892500901230478726142695401, 10.24929272984401266068861580133, 11.47971595988915079607491982129, 12.56364111102126933735376891264

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