Properties

Label 2-336-7.3-c6-0-7
Degree $2$
Conductor $336$
Sign $-0.944 - 0.327i$
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 + (27.1 − 15.6i)5-s + (342. − 21.6i)7-s + (121.5 + 210. i)9-s + (−526. + 912. i)11-s + 3.03e3i·13-s + 489.·15-s + (−4.83e3 − 2.79e3i)17-s + (−8.22e3 + 4.75e3i)19-s + (4.78e3 + 2.37e3i)21-s + (−1.15e4 − 1.99e4i)23-s + (−7.32e3 + 1.26e4i)25-s + 3.78e3i·27-s + 4.23e4·29-s + (−4.35e4 − 2.51e4i)31-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (0.217 − 0.125i)5-s + (0.998 − 0.0631i)7-s + (0.166 + 0.288i)9-s + (−0.395 + 0.685i)11-s + 1.38i·13-s + 0.144·15-s + (−0.983 − 0.568i)17-s + (−1.19 + 0.692i)19-s + (0.517 + 0.256i)21-s + (−0.947 − 1.64i)23-s + (−0.468 + 0.811i)25-s + 0.192i·27-s + 1.73·29-s + (−1.46 − 0.843i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.944 - 0.327i$
Analytic conductor: \(77.2981\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3),\ -0.944 - 0.327i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.137250034\)
\(L(\frac12)\) \(\approx\) \(1.137250034\)
\(L(4)\) 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 \( 1 + (-342. + 21.6i)T \)
good5 \( 1 + (-27.1 + 15.6i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (526. - 912. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 3.03e3iT - 4.82e6T^{2} \)
17 \( 1 + (4.83e3 + 2.79e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (8.22e3 - 4.75e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (1.15e4 + 1.99e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 4.23e4T + 5.94e8T^{2} \)
31 \( 1 + (4.35e4 + 2.51e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (1.21e4 + 2.09e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 1.34e5iT - 4.75e9T^{2} \)
43 \( 1 - 3.78e4T + 6.32e9T^{2} \)
47 \( 1 + (2.69e4 - 1.55e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-1.83e4 + 3.17e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-6.74e3 - 3.89e3i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.59e5 - 9.23e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.41e5 - 2.45e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 3.64e4T + 1.28e11T^{2} \)
73 \( 1 + (6.06e5 + 3.49e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-8.00e4 - 1.38e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 4.44e5iT - 3.26e11T^{2} \)
89 \( 1 + (6.10e5 - 3.52e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.33e5iT - 8.32e11T^{2} \)
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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

−10.87903013622599188885491709939, −9.996693654480873318764348587962, −8.992003213583796259275321309673, −8.286329023414729718203052187282, −7.24926304275242894476068946813, −6.15702757597371297904092601417, −4.59782495996952835555124755225, −4.29858558262966373223863724538, −2.39458375571346075597582544937, −1.71643712206865327308667858428, 0.21610767831528016357935567671, 1.66505149167892677930954589670, 2.66210759459276165913189991512, 3.94465296532921731932075514677, 5.22530830237491757942903469002, 6.19137614319541584213815583343, 7.45656847896733710030058619506, 8.308820366913634121682001268035, 8.859549043426861453096234032740, 10.36436875918173840724975039908

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