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Properties

Label 2-336-7.5-c6-0-29
Degree $2$
Conductor $336$
Sign $0.992 + 0.120i$
Analytic cond. $77.2981$
Root an. cond. $8.79193$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.5 − 7.79i)3-s + (76.6 + 44.2i)5-s + (298. + 169. i)7-s + (121.5 − 210. i)9-s + (−194. − 337. i)11-s + 157. i·13-s + 1.37e3·15-s + (4.44e3 − 2.56e3i)17-s + (−6.75 − 3.89i)19-s + (5.34e3 − 29.9i)21-s + (−80.9 + 140. i)23-s + (−3.90e3 − 6.75e3i)25-s − 3.78e3i·27-s + 2.66e4·29-s + (3.68e4 − 2.12e4i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.612 + 0.353i)5-s + (0.868 + 0.495i)7-s + (0.166 − 0.288i)9-s + (−0.146 − 0.253i)11-s + 0.0717i·13-s + 0.408·15-s + (0.904 − 0.522i)17-s + (−0.000984 − 0.000568i)19-s + (0.577 − 0.00323i)21-s + (−0.00665 + 0.0115i)23-s + (−0.249 − 0.432i)25-s − 0.192i·27-s + 1.09·29-s + (1.23 − 0.714i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.992 + 0.120i$
Analytic conductor: \(77.2981\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3),\ 0.992 + 0.120i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.576594136\)
\(L(\frac12)\) \(\approx\) \(3.576594136\)
\(L(4)\) 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 \( 1 + (-13.5 + 7.79i)T \)
7 \( 1 + (-298. - 169. i)T \)
good5 \( 1 + (-76.6 - 44.2i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (194. + 337. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 157. iT - 4.82e6T^{2} \)
17 \( 1 + (-4.44e3 + 2.56e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (6.75 + 3.89i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (80.9 - 140. i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 2.66e4T + 5.94e8T^{2} \)
31 \( 1 + (-3.68e4 + 2.12e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (4.85e3 - 8.41e3i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 4.65e4iT - 4.75e9T^{2} \)
43 \( 1 + 4.25e4T + 6.32e9T^{2} \)
47 \( 1 + (1.42e5 + 8.21e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-1.17e5 - 2.03e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (5.95e4 - 3.43e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-3.23e5 - 1.86e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.49e5 + 2.58e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 4.29e5T + 1.28e11T^{2} \)
73 \( 1 + (-3.87e4 + 2.23e4i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (8.65e4 - 1.49e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + 4.23e5iT - 3.26e11T^{2} \)
89 \( 1 + (-2.91e5 - 1.68e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 4.15e5iT - 8.32e11T^{2} \)
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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

−10.36237051365487949379114632554, −9.627740995930043173178056487486, −8.484830967900131067424704307173, −7.86864602232399581219804412809, −6.65756906341238246267317842708, −5.68376736177159825070797223330, −4.58485939458909359060289763247, −3.04661159300215705414026881000, −2.16200289176215233285385972416, −0.970079362254546482451232558005, 1.03123995261934713415546175299, 2.02472883675950075592790961483, 3.42077692367320257656462269479, 4.62980563939574744561768840510, 5.43267574109588434760796549992, 6.78409461273851242049628225798, 7.964616811729180368226275364195, 8.549310098761986092380075992434, 9.794197347320441886004328021211, 10.29370516197237504829884515002

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