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Properties

Label 2-336-21.17-c3-0-4
Degree $2$
Conductor $336$
Sign $-0.979 + 0.203i$
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  + (1.35 + 5.01i)3-s + (2.40 + 4.16i)5-s + (3.40 − 18.2i)7-s + (−23.3 + 13.5i)9-s + (−29.6 − 17.1i)11-s + 17.1i·13-s + (−17.6 + 17.6i)15-s + (−50.4 + 87.4i)17-s + (−119. + 69.2i)19-s + (95.9 − 7.51i)21-s + (−0.541 + 0.312i)23-s + (50.9 − 88.2i)25-s + (−99.5 − 98.8i)27-s + 222. i·29-s + (−247. − 143. i)31-s + ⋯
L(s)  = 1  + (0.259 + 0.965i)3-s + (0.215 + 0.372i)5-s + (0.183 − 0.982i)7-s + (−0.864 + 0.501i)9-s + (−0.813 − 0.469i)11-s + 0.366i·13-s + (−0.303 + 0.304i)15-s + (−0.720 + 1.24i)17-s + (−1.44 + 0.835i)19-s + (0.996 − 0.0780i)21-s + (−0.00490 + 0.00283i)23-s + (0.407 − 0.705i)25-s + (−0.709 − 0.704i)27-s + 1.42i·29-s + (−1.43 − 0.829i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.979 + 0.203i$
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: \((2,\ 336,\ (\ :3/2),\ -0.979 + 0.203i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5907968572\)
\(L(\frac12)\) \(\approx\) \(0.5907968572\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.35 - 5.01i)T \)
7 \( 1 + (-3.40 + 18.2i)T \)
good5 \( 1 + (-2.40 - 4.16i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (29.6 + 17.1i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 17.1iT - 2.19e3T^{2} \)
17 \( 1 + (50.4 - 87.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (119. - 69.2i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (0.541 - 0.312i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 222. iT - 2.43e4T^{2} \)
31 \( 1 + (247. + 143. i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-9.19 - 15.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 219.T + 6.89e4T^{2} \)
43 \( 1 - 370.T + 7.95e4T^{2} \)
47 \( 1 + (143. + 247. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-29.3 - 16.9i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-127. + 220. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (29.2 - 16.8i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (87.9 - 152. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 626. iT - 3.57e5T^{2} \)
73 \( 1 + (-189. - 109. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-263. - 456. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 370.T + 5.71e5T^{2} \)
89 \( 1 + (412. + 714. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.30e3iT - 9.12e5T^{2} \)
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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

−11.02354850082561611038244686418, −10.73000263078801391221608786973, −10.03300898151456992577039330472, −8.756865620650571507555686689069, −8.085718645898707844681585716946, −6.77278760401778240873746479620, −5.66196852151355266398327363906, −4.38675207889924573488451183049, −3.59923256696362669411928066697, −2.10888159642603298704476291706, 0.18386036817522194869431753679, 1.98623941903398390348662875037, 2.80312817685566352143735370942, 4.77958956489563528839390017388, 5.72062725960724178011569257467, 6.83283191852449328507024134945, 7.78512238827853282355678160590, 8.781753074690969752053918923453, 9.338656862968032769023698021534, 10.83411456241040006253465954964

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