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Properties

Label 2-336-21.17-c3-0-25
Degree $2$
Conductor $336$
Sign $-0.355 - 0.934i$
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  + (0.726 + 5.14i)3-s + (9.08 + 15.7i)5-s + (18.4 − 1.77i)7-s + (−25.9 + 7.47i)9-s + (57.7 + 33.3i)11-s + 21.6i·13-s + (−74.3 + 58.1i)15-s + (17.4 − 30.2i)17-s + (98.3 − 56.7i)19-s + (22.5 + 93.5i)21-s + (108. − 62.8i)23-s + (−102. + 177. i)25-s + (−57.3 − 128. i)27-s − 67.8i·29-s + (−211. − 122. i)31-s + ⋯
L(s)  = 1  + (0.139 + 0.990i)3-s + (0.812 + 1.40i)5-s + (0.995 − 0.0959i)7-s + (−0.960 + 0.276i)9-s + (1.58 + 0.914i)11-s + 0.461i·13-s + (−1.28 + 1.00i)15-s + (0.249 − 0.431i)17-s + (1.18 − 0.685i)19-s + (0.234 + 0.972i)21-s + (0.986 − 0.569i)23-s + (−0.820 + 1.42i)25-s + (−0.408 − 0.912i)27-s − 0.434i·29-s + (−1.22 − 0.708i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.355 - 0.934i$
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: \((2,\ 336,\ (\ :3/2),\ -0.355 - 0.934i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.739588235\)
\(L(\frac12)\) \(\approx\) \(2.739588235\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.726 - 5.14i)T \)
7 \( 1 + (-18.4 + 1.77i)T \)
good5 \( 1 + (-9.08 - 15.7i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-57.7 - 33.3i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 21.6iT - 2.19e3T^{2} \)
17 \( 1 + (-17.4 + 30.2i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-98.3 + 56.7i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-108. + 62.8i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 67.8iT - 2.43e4T^{2} \)
31 \( 1 + (211. + 122. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (37.1 + 64.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 302.T + 6.89e4T^{2} \)
43 \( 1 + 213.T + 7.95e4T^{2} \)
47 \( 1 + (-1.36 - 2.36i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-67.0 - 38.6i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (248. - 430. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (137. - 79.5i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-475. + 823. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 88.4iT - 3.57e5T^{2} \)
73 \( 1 + (582. + 336. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (198. + 343. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 151.T + 5.71e5T^{2} \)
89 \( 1 + (665. + 1.15e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 500. iT - 9.12e5T^{2} \)
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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

−11.31889048873004827110950705609, −10.45655465274512450694614821312, −9.567105517101503618703795855312, −9.014512929814322778001855275153, −7.42073972665684094766372621285, −6.65008019789386094838628313928, −5.38530721131100399847135127423, −4.32301113225649946061692979697, −3.10931360901666531032300920799, −1.83024068693414234679091864637, 1.19435528581147870044789611769, 1.49665896276066933121486043831, 3.48768030018479394592936523611, 5.20025928109054737921878450798, 5.73714307353452064179164756536, 7.00616205358565541710642985500, 8.274778730776220053663970479079, 8.737632319767386977828890245664, 9.592358238331674040494764313848, 11.14795701019442219313013733960

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