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Properties

Label 2-336-21.5-c3-0-40
Degree $2$
Conductor $336$
Sign $0.218 + 0.975i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.65 + 2.30i)3-s + (7.91 − 13.7i)5-s + (−13.3 − 12.8i)7-s + (16.3 + 21.4i)9-s + (−4.95 + 2.86i)11-s − 50.0i·13-s + (68.4 − 45.5i)15-s + (−39.4 − 68.2i)17-s + (81.4 + 47.0i)19-s + (−32.7 − 90.4i)21-s + (−96.5 − 55.7i)23-s + (−62.7 − 108. i)25-s + (26.5 + 137. i)27-s − 237. i·29-s + (−77.9 + 45.0i)31-s + ⋯
L(s)  = 1  + (0.895 + 0.444i)3-s + (0.707 − 1.22i)5-s + (−0.722 − 0.691i)7-s + (0.605 + 0.795i)9-s + (−0.135 + 0.0784i)11-s − 1.06i·13-s + (1.17 − 0.784i)15-s + (−0.562 − 0.973i)17-s + (0.983 + 0.568i)19-s + (−0.340 − 0.940i)21-s + (−0.875 − 0.505i)23-s + (−0.502 − 0.869i)25-s + (0.188 + 0.981i)27-s − 1.52i·29-s + (−0.451 + 0.260i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.218 + 0.975i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.218 + 0.975i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.427585351\)
\(L(\frac12)\) \(\approx\) \(2.427585351\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.65 - 2.30i)T \)
7 \( 1 + (13.3 + 12.8i)T \)
good5 \( 1 + (-7.91 + 13.7i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (4.95 - 2.86i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 50.0iT - 2.19e3T^{2} \)
17 \( 1 + (39.4 + 68.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-81.4 - 47.0i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (96.5 + 55.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 237. iT - 2.43e4T^{2} \)
31 \( 1 + (77.9 - 45.0i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (27.2 - 47.2i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 206.T + 6.89e4T^{2} \)
43 \( 1 - 507.T + 7.95e4T^{2} \)
47 \( 1 + (-53.6 + 92.8i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-373. + 215. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-212. - 367. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-397. - 229. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (476. + 825. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 7.43iT - 3.57e5T^{2} \)
73 \( 1 + (-812. + 469. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (454. - 787. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 199.T + 5.71e5T^{2} \)
89 \( 1 + (-571. + 989. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.26e3iT - 9.12e5T^{2} \)
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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

−10.50365314127505964819346439683, −9.829389130702968929671615937464, −9.223009279770536300108509924327, −8.219745249693605597065790221960, −7.33942850941327546813959441665, −5.81971577790446147642939459833, −4.81634685949169707503250547782, −3.71606744861125496693014507206, −2.38815491722884869055711479297, −0.76128614624686655105220827994, 1.87744530204955892177389394781, 2.76896480170700309647420504692, 3.80105384924613922716387301021, 5.75350369084181511530007487888, 6.65835884105395171343226100403, 7.28779759369534181073411746858, 8.700687622226467859405240998941, 9.398870060932505161521296429783, 10.20479452714262774437853047362, 11.29961953380213405912476634180

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