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Properties

Label 2-336-21.20-c1-0-8
Degree $2$
Conductor $336$
Sign $-0.350 + 0.936i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.66 + 0.468i)3-s − 0.936·5-s + (−1.56 + 2.13i)7-s + (2.56 − 1.56i)9-s − 5.12i·11-s − 3.33i·13-s + (1.56 − 0.438i)15-s − 4.27·17-s − 5.73i·19-s + (1.60 − 4.29i)21-s + 2i·23-s − 4.12·25-s + (−3.54 + 3.80i)27-s − 0.876i·29-s − 6.14i·31-s + ⋯
L(s)  = 1  + (−0.962 + 0.270i)3-s − 0.418·5-s + (−0.590 + 0.807i)7-s + (0.853 − 0.520i)9-s − 1.54i·11-s − 0.924i·13-s + (0.403 − 0.113i)15-s − 1.03·17-s − 1.31i·19-s + (0.350 − 0.936i)21-s + 0.417i·23-s − 0.824·25-s + (−0.681 + 0.731i)27-s − 0.162i·29-s − 1.10i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.350 + 0.936i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.350 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248176 - 0.357668i\)
\(L(\frac12)\) \(\approx\) \(0.248176 - 0.357668i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 - 0.468i)T \)
7 \( 1 + (1.56 - 2.13i)T \)
good5 \( 1 + 0.936T + 5T^{2} \)
11 \( 1 + 5.12iT - 11T^{2} \)
13 \( 1 + 3.33iT - 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
19 \( 1 + 5.73iT - 19T^{2} \)
23 \( 1 - 2iT - 23T^{2} \)
29 \( 1 + 0.876iT - 29T^{2} \)
31 \( 1 + 6.14iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 0.525T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 0.876iT - 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 13.1iT - 71T^{2} \)
73 \( 1 + 3.74iT - 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 + 3.33T + 83T^{2} \)
89 \( 1 - 15.7T + 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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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

−11.30713005049328599916631116650, −10.60004236166578583956985073344, −9.398304647832427662803812011711, −8.615221690366633421121445953205, −7.31729778640392544401321399703, −6.09673460847483120623364958791, −5.57757633980518531089908001226, −4.18698246745623218711546618449, −2.89131311290160942616358338992, −0.33789138564938135690419200069, 1.79930769872408241502212414992, 3.99105783358831616684789652388, 4.70210203910250901032873659074, 6.21783067904253449024016031084, 7.00388490318146322968631054988, 7.68715247828137325168199540250, 9.251806751584052602505561735920, 10.20691952364794359723587717854, 10.84962279534311731894749905230, 12.05690521155844526205046364606

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