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Properties

Label 2-336-7.2-c3-0-3
Degree $2$
Conductor $336$
Sign $-0.940 + 0.340i$
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  + (1.5 + 2.59i)3-s + (−9.47 + 16.4i)5-s + (−12.8 + 13.3i)7-s + (−4.5 + 7.79i)9-s + (27.3 + 47.3i)11-s + 62.0·13-s − 56.8·15-s + (−61.2 − 106. i)17-s + (−6.25 + 10.8i)19-s + (−53.9 − 13.1i)21-s + (−37.2 + 64.4i)23-s + (−117. − 203. i)25-s − 27·27-s − 232.·29-s + (5.18 + 8.97i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.847 + 1.46i)5-s + (−0.691 + 0.722i)7-s + (−0.166 + 0.288i)9-s + (0.750 + 1.29i)11-s + 1.32·13-s − 0.978·15-s + (−0.873 − 1.51i)17-s + (−0.0755 + 0.130i)19-s + (−0.560 − 0.137i)21-s + (−0.337 + 0.584i)23-s + (−0.937 − 1.62i)25-s − 0.192·27-s − 1.48·29-s + (0.0300 + 0.0520i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.940 + 0.340i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.940 + 0.340i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.940 + 0.340i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.029642017\)
\(L(\frac12)\) \(\approx\) \(1.029642017\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.5 - 2.59i)T \)
7 \( 1 + (12.8 - 13.3i)T \)
good5 \( 1 + (9.47 - 16.4i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-27.3 - 47.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 62.0T + 2.19e3T^{2} \)
17 \( 1 + (61.2 + 106. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (6.25 - 10.8i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (37.2 - 64.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 232.T + 2.43e4T^{2} \)
31 \( 1 + (-5.18 - 8.97i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-122. + 213. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 238.T + 6.89e4T^{2} \)
43 \( 1 - 92.9T + 7.95e4T^{2} \)
47 \( 1 + (242. - 420. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-189. - 327. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (91.3 + 158. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (198. - 343. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (130. + 226. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 874.T + 3.57e5T^{2} \)
73 \( 1 + (76.2 + 131. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-286. + 496. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 317.T + 5.71e5T^{2} \)
89 \( 1 + (-47.5 + 82.2i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.60e3T + 9.12e5T^{2} \)
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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

−11.38826809425070170507905901892, −10.91185573303845947588786009699, −9.617737512411200028718128282139, −9.135857917712579334703093230202, −7.68754245224527627180621449201, −6.91971893673100177999294184930, −5.97208705580859803984144980312, −4.28146364152356931814644105485, −3.42754701407145347667151440921, −2.35026848862183365049855170641, 0.37486828918244038711921786449, 1.37995615840591643674621248264, 3.65096517188406636216653849729, 4.10565331721750020041029153068, 5.84304865539605993223864103491, 6.66993750807694256302618602468, 8.136975472134210615120981863888, 8.512240637506422970484710254342, 9.341515558210572188976874567194, 10.86723571459848927478404631787

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