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Properties

Label 2-336-21.5-c3-0-10
Degree $2$
Conductor $336$
Sign $0.815 - 0.578i$
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  + (0.328 − 5.18i)3-s + (−7.91 + 13.7i)5-s + (−13.3 − 12.8i)7-s + (−26.7 − 3.41i)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 + (−70.8 + 65.1i)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  + (0.0633 − 0.997i)3-s + (−0.707 + 1.22i)5-s + (−0.722 − 0.691i)7-s + (−0.991 − 0.126i)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 + (−0.735 + 0.677i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.815 - 0.578i$
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: \((2,\ 336,\ (\ :3/2),\ 0.815 - 0.578i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.182898212\)
\(L(\frac12)\) \(\approx\) \(1.182898212\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.328 + 5.18i)T \)
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

−11.01213455568134158635716403558, −10.66097961325419154203387136007, −9.366223689953010850397356529997, −7.923729279393573359184898401798, −7.43415000289300838001121065759, −6.61485564408433524466511129454, −5.61159777727367852455349772350, −3.53057487088247624846451541709, −3.04434508580031457039084140678, −1.07990150874983225885306005109, 0.51398074611760019973483438256, 2.74628912964605480651357879804, 4.04836657196565785352763528838, 4.87601996915776190366502008286, 5.83793976376522953347107274072, 7.32810358182824059944425336656, 8.563147538002910799148512022747, 9.280529338683612751878750974483, 9.691456970465644051082461729386, 11.26992560354986383440641144192

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