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Properties

Label 2-336-21.17-c3-0-16
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: $\chi_{336} (17, \cdot )$
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

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

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