Properties

Label 2-336-7.5-c6-0-30
Degree $2$
Conductor $336$
Sign $0.686 + 0.727i$
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 + (44.9 + 25.9i)5-s + (120. + 321. i)7-s + (121.5 − 210. i)9-s + (−453. − 785. i)11-s − 510. i·13-s + 809.·15-s + (−566. + 327. i)17-s + (−2.88e3 − 1.66e3i)19-s + (4.13e3 + 3.39e3i)21-s + (9.70e3 − 1.68e4i)23-s + (−6.46e3 − 1.11e4i)25-s − 3.78e3i·27-s + 6.03e3·29-s + (−8.30e3 + 4.79e3i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.359 + 0.207i)5-s + (0.352 + 0.935i)7-s + (0.166 − 0.288i)9-s + (−0.340 − 0.590i)11-s − 0.232i·13-s + 0.239·15-s + (−0.115 + 0.0666i)17-s + (−0.420 − 0.242i)19-s + (0.446 + 0.366i)21-s + (0.797 − 1.38i)23-s + (−0.413 − 0.716i)25-s − 0.192i·27-s + 0.247·29-s + (−0.278 + 0.160i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.686 + 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.686 + 0.727i$
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: \((2,\ 336,\ (\ :3),\ 0.686 + 0.727i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.740117633\)
\(L(\frac12)\) \(\approx\) \(2.740117633\)
\(L(4)\) 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 \( 1 + (-120. - 321. i)T \)
good5 \( 1 + (-44.9 - 25.9i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (453. + 785. i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 510. iT - 4.82e6T^{2} \)
17 \( 1 + (566. - 327. i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (2.88e3 + 1.66e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-9.70e3 + 1.68e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 6.03e3T + 5.94e8T^{2} \)
31 \( 1 + (8.30e3 - 4.79e3i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-2.89e4 + 5.01e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 4.30e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.25e5T + 6.32e9T^{2} \)
47 \( 1 + (-1.76e5 - 1.01e5i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (4.26e4 + 7.37e4i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-2.08e5 + 1.20e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (1.85e5 + 1.06e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-1.46e5 - 2.53e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 6.22e4T + 1.28e11T^{2} \)
73 \( 1 + (9.70e4 - 5.60e4i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-3.77e5 + 6.54e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 9.10e4iT - 3.26e11T^{2} \)
89 \( 1 + (5.01e5 + 2.89e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 1.52e5iT - 8.32e11T^{2} \)
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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

−10.48006490680997584816752493346, −9.250368787573007983795727351418, −8.593206366208987074913068401727, −7.72606399892841875477138967648, −6.46870009858502424812845912665, −5.65185284917344568792411093285, −4.39348514337082035444664554925, −2.85141723512364117705107652468, −2.20002014452831837892392668107, −0.65729820613407984486957917116, 1.08511488486799431367275422982, 2.21653202491271264589236806145, 3.63840994992874162014850783797, 4.57367942028304611824732441009, 5.62467292374443269461153005130, 7.09990798800779904755509928816, 7.72421134355124174249961320110, 8.913997189408460567446893800276, 9.707334678190781709753334542100, 10.54421962504838632836143623001

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