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Properties

Label 2-336-7.2-c5-0-23
Degree $2$
Conductor $336$
Sign $0.657 + 0.753i$
Analytic cond. $53.8889$
Root an. cond. $7.34091$
Motivic weight $5$
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 + 7.79i)3-s + (−46.5 + 80.6i)5-s + (−119. − 49.6i)7-s + (−40.5 + 70.1i)9-s + (−50.2 − 87.1i)11-s − 333.·13-s − 837.·15-s + (34.0 + 58.9i)17-s + (−743. + 1.28e3i)19-s + (−151. − 1.15e3i)21-s + (800. − 1.38e3i)23-s + (−2.76e3 − 4.79e3i)25-s − 729·27-s − 6.73e3·29-s + (1.61e3 + 2.80e3i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.832 + 1.44i)5-s + (−0.923 − 0.383i)7-s + (−0.166 + 0.288i)9-s + (−0.125 − 0.217i)11-s − 0.547·13-s − 0.961·15-s + (0.0285 + 0.0494i)17-s + (−0.472 + 0.817i)19-s + (−0.0749 − 0.572i)21-s + (0.315 − 0.546i)23-s + (−0.886 − 1.53i)25-s − 0.192·27-s − 1.48·29-s + (0.302 + 0.523i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.657 + 0.753i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ 0.657 + 0.753i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4737346379\)
\(L(\frac12)\) \(\approx\) \(0.4737346379\)
\(L(\frac{7}{2})\) 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 \( 1 + (-4.5 - 7.79i)T \)
7 \( 1 + (119. + 49.6i)T \)
good5 \( 1 + (46.5 - 80.6i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (50.2 + 87.1i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 333.T + 3.71e5T^{2} \)
17 \( 1 + (-34.0 - 58.9i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (743. - 1.28e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-800. + 1.38e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 6.73e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.61e3 - 2.80e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (1.10e3 - 1.91e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 6.48e3T + 1.15e8T^{2} \)
43 \( 1 - 1.38e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.33e4 + 2.31e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (1.24e4 + 2.15e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-1.70e4 - 2.96e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (9.27e3 - 1.60e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-3.59e4 - 6.21e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 1.50e4T + 1.80e9T^{2} \)
73 \( 1 + (2.78e4 + 4.82e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-9.64e3 + 1.67e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 9.25e4T + 3.93e9T^{2} \)
89 \( 1 + (-3.58e4 + 6.21e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 1.45e5T + 8.58e9T^{2} \)
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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

−10.49046677245200757534398498711, −9.963518529072143881720699358523, −8.738230281892676008990974803445, −7.56237247686059573705785817346, −6.93150526988562787314124994167, −5.82765182167653650434655940214, −4.15495816669768855735282476665, −3.42628597272419629363304735839, −2.49358620303178510692736724332, −0.15442350486501182768417752402, 0.840606031507154982724676947388, 2.37388104348029120520237264490, 3.75017178909430232296744239333, 4.83619817453811958644224867214, 5.94367564749824308243691604974, 7.25986417712454229495103492176, 7.979608923686554498677131012469, 9.158629947157007351923772906441, 9.392448315057756720940223436195, 11.06028184838317158919893534691

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