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Properties

Label 2-336-21.5-c3-0-17
Degree $2$
Conductor $336$
Sign $0.406 - 0.913i$
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  + (0.0354 + 5.19i)3-s + (5.27 − 9.13i)5-s + (−17.7 − 5.44i)7-s + (−26.9 + 0.368i)9-s + (26.6 − 15.4i)11-s + 19.8i·13-s + (47.6 + 27.0i)15-s + (46.3 + 80.2i)17-s + (118. + 68.6i)19-s + (27.6 − 92.1i)21-s + (37.6 + 21.7i)23-s + (6.89 + 11.9i)25-s + (−2.87 − 140. i)27-s + 134. i·29-s + (−144. + 83.6i)31-s + ⋯
L(s)  = 1  + (0.00682 + 0.999i)3-s + (0.471 − 0.816i)5-s + (−0.955 − 0.293i)7-s + (−0.999 + 0.0136i)9-s + (0.731 − 0.422i)11-s + 0.423i·13-s + (0.820 + 0.466i)15-s + (0.661 + 1.14i)17-s + (1.43 + 0.828i)19-s + (0.287 − 0.957i)21-s + (0.341 + 0.197i)23-s + (0.0551 + 0.0956i)25-s + (−0.0204 − 0.999i)27-s + 0.860i·29-s + (−0.839 + 0.484i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.406 - 0.913i$
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: \((2,\ 336,\ (\ :3/2),\ 0.406 - 0.913i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.780648634\)
\(L(\frac12)\) \(\approx\) \(1.780648634\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.0354 - 5.19i)T \)
7 \( 1 + (17.7 + 5.44i)T \)
good5 \( 1 + (-5.27 + 9.13i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-26.6 + 15.4i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 19.8iT - 2.19e3T^{2} \)
17 \( 1 + (-46.3 - 80.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-118. - 68.6i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-37.6 - 21.7i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 134. iT - 2.43e4T^{2} \)
31 \( 1 + (144. - 83.6i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-191. + 332. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 107.T + 6.89e4T^{2} \)
43 \( 1 - 285.T + 7.95e4T^{2} \)
47 \( 1 + (120. - 209. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (432. - 249. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-366. - 634. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (265. + 153. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-280. - 485. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 74.2iT - 3.57e5T^{2} \)
73 \( 1 + (-141. + 81.6i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-437. + 757. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 406.T + 5.71e5T^{2} \)
89 \( 1 + (-526. + 911. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 243. iT - 9.12e5T^{2} \)
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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

−11.12979854085635402086844934264, −10.22237752625142989746842195319, −9.293423428843337109242214241903, −9.004556561792258435074406609261, −7.57258107044762490917336586963, −6.08801466865159570263106612435, −5.44166227899748961657700428745, −4.07116914384273184078697602375, −3.26163434776873663071890569088, −1.20015177112299287453344354577, 0.75005587393383002157546071393, 2.47725744540049495171215472103, 3.25146072374519910104003897649, 5.27681330481832406439624493235, 6.35018061819249684619108003086, 6.95420773756858975537512613145, 7.85335619538606178643818203419, 9.337480462104113324828604007758, 9.774343961367495060831123525679, 11.18047124550834956842142000353

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