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Properties

Label 2-336-21.17-c3-0-42
Degree $2$
Conductor $336$
Sign $-0.821 + 0.569i$
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  + (4.81 − 1.94i)3-s + (−9.08 − 15.7i)5-s + (18.4 − 1.77i)7-s + (19.4 − 18.7i)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 + (85.3 − 44.3i)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  + (0.927 − 0.373i)3-s + (−0.812 − 1.40i)5-s + (0.995 − 0.0959i)7-s + (0.720 − 0.693i)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 + (0.887 − 0.461i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.821 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.821 + 0.569i$
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: \((2,\ 336,\ (\ :3/2),\ -0.821 + 0.569i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.803006305\)
\(L(\frac12)\) \(\approx\) \(1.803006305\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.81 + 1.94i)T \)
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

−10.92846777041984411488220288713, −9.478806939347683484814461661612, −8.640101018573701776462620116143, −7.963302368650134597086114471363, −7.47981706318302474352504793201, −5.55715482699463270249569709151, −4.63047484745653908118772764315, −3.50397066306024772214869840405, −1.90331896470681911150760310578, −0.55570136832459955155357110986, 2.19364136473939826842579447435, 3.08412783356299449182028125409, 4.26835648450694257595906723190, 5.39616696469694345822384906567, 7.28129438732516483837159829075, 7.62438716843898695990996809397, 8.403418126518184505291044392166, 9.927942977997402848312070501988, 10.46173301971264980478715843140, 11.28349231512144660841461166221

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