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Properties

Label 2-336-7.4-c5-0-39
Degree $2$
Conductor $336$
Sign $-0.643 - 0.765i$
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 + (−11.8 − 20.5i)5-s + (−30.6 − 125. i)7-s + (−40.5 − 70.1i)9-s + (232. − 403. i)11-s − 1.01e3·13-s − 213.·15-s + (−280. + 486. i)17-s + (−693. − 1.20e3i)19-s + (−1.11e3 − 327. i)21-s + (2.05e3 + 3.56e3i)23-s + (1.28e3 − 2.21e3i)25-s − 729·27-s − 2.38e3·29-s + (1.47e3 − 2.55e3i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.212 − 0.367i)5-s + (−0.236 − 0.971i)7-s + (−0.166 − 0.288i)9-s + (0.580 − 1.00i)11-s − 1.67·13-s − 0.245·15-s + (−0.235 + 0.408i)17-s + (−0.440 − 0.763i)19-s + (−0.554 − 0.162i)21-s + (0.810 + 1.40i)23-s + (0.409 − 0.709i)25-s − 0.192·27-s − 0.525·29-s + (0.275 − 0.477i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.643 - 0.765i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ -0.643 - 0.765i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.4975474005\)
\(L(\frac12)\) \(\approx\) \(0.4975474005\)
\(L(\frac{7}{2})\) 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 \( 1 + (30.6 + 125. i)T \)
good5 \( 1 + (11.8 + 20.5i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-232. + 403. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 + 1.01e3T + 3.71e5T^{2} \)
17 \( 1 + (280. - 486. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (693. + 1.20e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-2.05e3 - 3.56e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 2.38e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.47e3 + 2.55e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (-4.95e3 - 8.58e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 4.47e3T + 1.15e8T^{2} \)
43 \( 1 + 5.18e3T + 1.47e8T^{2} \)
47 \( 1 + (1.56e3 + 2.70e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (570. - 988. i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-1.37e4 + 2.38e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.05e4 - 1.82e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.77e4 - 4.81e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 - 6.07e3T + 1.80e9T^{2} \)
73 \( 1 + (-8.38e3 + 1.45e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (2.42e3 + 4.19e3i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.01e4T + 3.93e9T^{2} \)
89 \( 1 + (-3.12e4 - 5.41e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + 6.36e4T + 8.58e9T^{2} \)
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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

−10.03790730907623829885694420740, −9.157188262082962667573157969372, −8.163126898505188046076987800341, −7.24876019980684960179307935100, −6.47445657731515549575054633156, −5.05093514403896208648678679816, −3.92784768087516316022765221630, −2.74058972691527770355361610593, −1.18554491418263021655227918587, −0.12705840911659962893721265145, 2.07499459641304160013335754294, 2.96865418977090041309928911677, 4.38067962247353719671097656997, 5.24115913523315616862724932723, 6.63157491942166852027754189527, 7.46609672419689305340875415716, 8.717901393692023178442625359039, 9.484434369130885819835879111270, 10.20373387846053485779911258342, 11.31380262956587091062706087629

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