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Properties

Label 2-336-16.5-c1-0-2
Degree $2$
Conductor $336$
Sign $0.510 - 0.860i$
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  + (−1.14 − 0.827i)2-s + (−0.707 + 0.707i)3-s + (0.631 + 1.89i)4-s + (2.39 + 2.39i)5-s + (1.39 − 0.226i)6-s − i·7-s + (0.844 − 2.69i)8-s − 1.00i·9-s + (−0.767 − 4.73i)10-s + (0.853 + 0.853i)11-s + (−1.78 − 0.895i)12-s + (−3.62 + 3.62i)13-s + (−0.827 + 1.14i)14-s − 3.39·15-s + (−3.20 + 2.39i)16-s + 5.93·17-s + ⋯
L(s)  = 1  + (−0.811 − 0.584i)2-s + (−0.408 + 0.408i)3-s + (0.315 + 0.948i)4-s + (1.07 + 1.07i)5-s + (0.569 − 0.0923i)6-s − 0.377i·7-s + (0.298 − 0.954i)8-s − 0.333i·9-s + (−0.242 − 1.49i)10-s + (0.257 + 0.257i)11-s + (−0.516 − 0.258i)12-s + (−1.00 + 1.00i)13-s + (−0.221 + 0.306i)14-s − 0.875·15-s + (−0.800 + 0.599i)16-s + 1.44·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.510 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.510 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.510 - 0.860i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.510 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.766383 + 0.436465i\)
\(L(\frac12)\) \(\approx\) \(0.766383 + 0.436465i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 + 0.827i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + iT \)
good5 \( 1 + (-2.39 - 2.39i)T + 5iT^{2} \)
11 \( 1 + (-0.853 - 0.853i)T + 11iT^{2} \)
13 \( 1 + (3.62 - 3.62i)T - 13iT^{2} \)
17 \( 1 - 5.93T + 17T^{2} \)
19 \( 1 + (4.47 - 4.47i)T - 19iT^{2} \)
23 \( 1 + 2.09iT - 23T^{2} \)
29 \( 1 + (-0.577 + 0.577i)T - 29iT^{2} \)
31 \( 1 - 3.23T + 31T^{2} \)
37 \( 1 + (-6.16 - 6.16i)T + 37iT^{2} \)
41 \( 1 - 8.30iT - 41T^{2} \)
43 \( 1 + (5.68 + 5.68i)T + 43iT^{2} \)
47 \( 1 + 4.36T + 47T^{2} \)
53 \( 1 + (1.19 + 1.19i)T + 53iT^{2} \)
59 \( 1 + (-7.94 - 7.94i)T + 59iT^{2} \)
61 \( 1 + (5.56 - 5.56i)T - 61iT^{2} \)
67 \( 1 + (-7.32 + 7.32i)T - 67iT^{2} \)
71 \( 1 - 1.22iT - 71T^{2} \)
73 \( 1 + 16.7iT - 73T^{2} \)
79 \( 1 + 5.39T + 79T^{2} \)
83 \( 1 + (-9.38 + 9.38i)T - 83iT^{2} \)
89 \( 1 + 8.85iT - 89T^{2} \)
97 \( 1 - 14.4T + 97T^{2} \)
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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

−11.56332090212439986112852522112, −10.33086685851760279590194783692, −10.14126182496765074072632087568, −9.390385921677481062832747379609, −7.993998465583406134098563563770, −6.86989083525650624164164831912, −6.16713747053343446978440505740, −4.48126120717645192115782821819, −3.11220221360853471192335342234, −1.81910017400649468040993415387, 0.883119955549150779656273449301, 2.32444482931548686509717082903, 5.04039399144946311994696600595, 5.54979628730903561526334270392, 6.50397606494877416827092996744, 7.73034802768833264940756916628, 8.574009688446250055370582208176, 9.535427519550073142662730812012, 10.12217820769156992647374166062, 11.28677692613218246910262722089

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