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Properties

Label 2-336-336.173-c1-0-55
Degree $2$
Conductor $336$
Sign $0.528 + 0.849i$
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  + (1.40 − 0.131i)2-s + (1.11 − 1.32i)3-s + (1.96 − 0.369i)4-s + (−0.260 − 0.973i)5-s + (1.39 − 2.01i)6-s + (−2.63 + 0.251i)7-s + (2.71 − 0.777i)8-s + (−0.512 − 2.95i)9-s + (−0.494 − 1.33i)10-s + (0.123 − 0.460i)11-s + (1.70 − 3.01i)12-s + (−2.21 + 2.21i)13-s + (−3.67 + 0.699i)14-s + (−1.58 − 0.739i)15-s + (3.72 − 1.45i)16-s + (2.95 + 5.11i)17-s + ⋯
L(s)  = 1  + (0.995 − 0.0926i)2-s + (0.643 − 0.765i)3-s + (0.982 − 0.184i)4-s + (−0.116 − 0.435i)5-s + (0.570 − 0.821i)6-s + (−0.995 + 0.0950i)7-s + (0.961 − 0.274i)8-s + (−0.170 − 0.985i)9-s + (−0.156 − 0.422i)10-s + (0.0371 − 0.138i)11-s + (0.491 − 0.870i)12-s + (−0.613 + 0.613i)13-s + (−0.982 + 0.186i)14-s + (−0.408 − 0.191i)15-s + (0.931 − 0.362i)16-s + (0.716 + 1.24i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.528 + 0.849i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.528 + 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29455 - 1.27509i\)
\(L(\frac12)\) \(\approx\) \(2.29455 - 1.27509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.131i)T \)
3 \( 1 + (-1.11 + 1.32i)T \)
7 \( 1 + (2.63 - 0.251i)T \)
good5 \( 1 + (0.260 + 0.973i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.123 + 0.460i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (2.21 - 2.21i)T - 13iT^{2} \)
17 \( 1 + (-2.95 - 5.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.66 - 6.22i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.412 + 0.714i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.21 + 1.21i)T - 29iT^{2} \)
31 \( 1 + (5.94 - 3.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.872 + 3.25i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.00iT - 41T^{2} \)
43 \( 1 + (5.59 - 5.59i)T - 43iT^{2} \)
47 \( 1 + (-5.11 + 8.86i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.25 - 4.68i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (12.9 + 3.46i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.511 - 1.90i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.00 + 11.2i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.11T + 71T^{2} \)
73 \( 1 + (-2.26 - 3.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.39 - 14.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.39 + 9.39i)T + 83iT^{2} \)
89 \( 1 + (-5.39 - 3.11i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.65iT - 97T^{2} \)
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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

−12.04788890996264923752681050778, −10.54125910584137980892011820997, −9.586875662619095025019843128667, −8.421300191754238458268699191866, −7.40337444763967873876332224100, −6.47515091730091840383466884624, −5.61294369603350136272615778708, −4.01060008718570277092017034386, −3.11660275048290363461166947490, −1.66854955537877786474344907200, 2.79092939749654030422431591106, 3.24593860400844940892255477521, 4.63611604793408494190935168248, 5.53288563139174432749954034469, 6.99530936534534729922205622463, 7.55500028145174346936335995387, 9.098759259549692960882787685484, 9.945498837042056555603272150899, 10.80505304826793160940473664036, 11.72882207998660376754048790825

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