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Properties

Label 2-336-12.11-c3-0-29
Degree $2$
Conductor $336$
Sign $-0.100 + 0.994i$
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  + (3.03 − 4.21i)3-s + 3.22i·5-s + 7i·7-s + (−8.57 − 25.6i)9-s − 9.66·11-s + 34.8·13-s + (13.5 + 9.78i)15-s − 129. i·17-s − 67.0i·19-s + (29.5 + 21.2i)21-s + 15.6·23-s + 114.·25-s + (−133. − 41.5i)27-s − 87.6i·29-s − 143. i·31-s + ⋯
L(s)  = 1  + (0.584 − 0.811i)3-s + 0.288i·5-s + 0.377i·7-s + (−0.317 − 0.948i)9-s − 0.264·11-s + 0.743·13-s + (0.234 + 0.168i)15-s − 1.84i·17-s − 0.809i·19-s + (0.306 + 0.220i)21-s + 0.141·23-s + 0.916·25-s + (−0.955 − 0.296i)27-s − 0.561i·29-s − 0.831i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.100 + 0.994i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ -0.100 + 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.036590699\)
\(L(\frac12)\) \(\approx\) \(2.036590699\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.03 + 4.21i)T \)
7 \( 1 - 7iT \)
good5 \( 1 - 3.22iT - 125T^{2} \)
11 \( 1 + 9.66T + 1.33e3T^{2} \)
13 \( 1 - 34.8T + 2.19e3T^{2} \)
17 \( 1 + 129. iT - 4.91e3T^{2} \)
19 \( 1 + 67.0iT - 6.85e3T^{2} \)
23 \( 1 - 15.6T + 1.21e4T^{2} \)
29 \( 1 + 87.6iT - 2.43e4T^{2} \)
31 \( 1 + 143. iT - 2.97e4T^{2} \)
37 \( 1 - 104.T + 5.06e4T^{2} \)
41 \( 1 - 257. iT - 6.89e4T^{2} \)
43 \( 1 + 267. iT - 7.95e4T^{2} \)
47 \( 1 + 430.T + 1.03e5T^{2} \)
53 \( 1 + 121. iT - 1.48e5T^{2} \)
59 \( 1 + 861.T + 2.05e5T^{2} \)
61 \( 1 - 502.T + 2.26e5T^{2} \)
67 \( 1 - 162. iT - 3.00e5T^{2} \)
71 \( 1 - 616.T + 3.57e5T^{2} \)
73 \( 1 - 719.T + 3.89e5T^{2} \)
79 \( 1 + 250. iT - 4.93e5T^{2} \)
83 \( 1 - 376.T + 5.71e5T^{2} \)
89 \( 1 + 870. iT - 7.04e5T^{2} \)
97 \( 1 - 44.3T + 9.12e5T^{2} \)
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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

−11.15622360823790645411015482817, −9.700030206779051814256641100711, −8.966245023046935357045068455746, −7.991626539439038045157299420607, −7.07304787699756473388668887539, −6.22791599120575431910132042562, −4.90174899523329734643573928494, −3.25369299542951427504279573491, −2.35775230286848540201520776046, −0.69872570312885537254986868744, 1.57866321928630533363938601654, 3.26857287606586908799155691294, 4.14634682589570850415623512918, 5.28056460855044436194971039976, 6.46641303293653855486483365598, 7.965824856483088516173689204643, 8.505499028177168418445171646416, 9.507435030693746576171611132086, 10.57468167860345923701182584856, 10.91107676053688654114286861049

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