Properties

Label 2-336-21.5-c3-0-25
Degree $2$
Conductor $336$
Sign $0.683 - 0.729i$
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  + (2.82 + 4.36i)3-s + (9.22 − 15.9i)5-s + (6.70 + 17.2i)7-s + (−11.0 + 24.6i)9-s + (−11.8 + 6.86i)11-s − 9.57i·13-s + (95.7 − 4.87i)15-s + (48.2 + 83.5i)17-s + (23.2 + 13.4i)19-s + (−56.3 + 77.9i)21-s + (153. + 88.8i)23-s + (−107. − 186. i)25-s + (−138. + 21.2i)27-s − 131. i·29-s + (273. − 157. i)31-s + ⋯
L(s)  = 1  + (0.543 + 0.839i)3-s + (0.825 − 1.42i)5-s + (0.361 + 0.932i)7-s + (−0.409 + 0.912i)9-s + (−0.326 + 0.188i)11-s − 0.204i·13-s + (1.64 − 0.0838i)15-s + (0.688 + 1.19i)17-s + (0.280 + 0.162i)19-s + (−0.585 + 0.810i)21-s + (1.39 + 0.805i)23-s + (−0.862 − 1.49i)25-s + (−0.988 + 0.151i)27-s − 0.840i·29-s + (1.58 − 0.914i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.683 - 0.729i$
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: \((2,\ 336,\ (\ :3/2),\ 0.683 - 0.729i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.713571698\)
\(L(\frac12)\) \(\approx\) \(2.713571698\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.82 - 4.36i)T \)
7 \( 1 + (-6.70 - 17.2i)T \)
good5 \( 1 + (-9.22 + 15.9i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (11.8 - 6.86i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 9.57iT - 2.19e3T^{2} \)
17 \( 1 + (-48.2 - 83.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-23.2 - 13.4i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-153. - 88.8i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 131. iT - 2.43e4T^{2} \)
31 \( 1 + (-273. + 157. i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (54.7 - 94.7i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 390.T + 6.89e4T^{2} \)
43 \( 1 + 291.T + 7.95e4T^{2} \)
47 \( 1 + (149. - 258. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-6.68 + 3.85i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-133. - 231. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-195. - 113. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (296. + 512. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 864. iT - 3.57e5T^{2} \)
73 \( 1 + (-474. + 274. i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (91.1 - 157. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 54.3T + 5.71e5T^{2} \)
89 \( 1 + (403. - 698. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.42e3iT - 9.12e5T^{2} \)
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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

−11.18695063871534345763493478599, −9.917915328037678791542128286215, −9.470273445118542219995150002986, −8.495533031555214072581472677242, −7.992305537805278555161360119569, −5.88679696392872617300063837758, −5.26321482722551291673834689665, −4.35250961792797111159481940066, −2.74925632190122225648248981286, −1.44578845149932664990505191359, 1.03915890268930721938987421686, 2.54925029059186940707667988446, 3.30998525525235686668836525412, 5.14559283284370040598000672710, 6.62022540326521010734560808985, 6.99053680423672259874771102643, 7.923161075117070889695467241834, 9.176394538023806146301827037285, 10.17709877508762452979297936323, 10.91200758341308622249721306849

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