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Properties

Label 2-336-7.4-c5-0-34
Degree $2$
Conductor $336$
Sign $-0.988 - 0.150i$
Analytic cond. $53.8889$
Root an. cond. $7.34091$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.5 + 7.79i)3-s + (−19.4 − 33.7i)5-s + (−106. − 74.3i)7-s + (−40.5 − 70.1i)9-s + (47.5 − 82.4i)11-s + 869.·13-s + 350.·15-s + (311. − 539. i)17-s + (−447. − 775. i)19-s + (1.05e3 − 492. i)21-s + (811. + 1.40e3i)23-s + (805. − 1.39e3i)25-s + 729·27-s − 2.92e3·29-s + (−4.04e3 + 6.99e3i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.348 − 0.602i)5-s + (−0.819 − 0.573i)7-s + (−0.166 − 0.288i)9-s + (0.118 − 0.205i)11-s + 1.42·13-s + 0.401·15-s + (0.261 − 0.452i)17-s + (−0.284 − 0.492i)19-s + (0.523 − 0.243i)21-s + (0.319 + 0.554i)23-s + (0.257 − 0.446i)25-s + 0.192·27-s − 0.645·29-s + (−0.755 + 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ -0.988 - 0.150i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1674203848\)
\(L(\frac12)\) \(\approx\) \(0.1674203848\)
\(L(\frac{7}{2})\) 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 \( 1 + (4.5 - 7.79i)T \)
7 \( 1 + (106. + 74.3i)T \)
good5 \( 1 + (19.4 + 33.7i)T + (-1.56e3 + 2.70e3i)T^{2} \)
11 \( 1 + (-47.5 + 82.4i)T + (-8.05e4 - 1.39e5i)T^{2} \)
13 \( 1 - 869.T + 3.71e5T^{2} \)
17 \( 1 + (-311. + 539. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (447. + 775. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-811. - 1.40e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + 2.92e3T + 2.05e7T^{2} \)
31 \( 1 + (4.04e3 - 6.99e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (2.72e3 + 4.72e3i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 + 2.35e3T + 1.15e8T^{2} \)
43 \( 1 + 2.40e3T + 1.47e8T^{2} \)
47 \( 1 + (1.13e3 + 1.96e3i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-1.24e4 + 2.14e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (1.21e4 - 2.11e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (2.00e4 + 3.46e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (2.37e4 - 4.12e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + 4.60e4T + 1.80e9T^{2} \)
73 \( 1 + (3.68e3 - 6.38e3i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (3.35e4 + 5.81e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 6.88e3T + 3.93e9T^{2} \)
89 \( 1 + (-6.43e3 - 1.11e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 - 1.98e4T + 8.58e9T^{2} \)
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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

−10.35176434183142218857401345377, −9.210102021735026233797400825845, −8.616119929717663177818611257596, −7.28018948052116530764264304386, −6.28445555791230107334439589402, −5.21087305619787048963898794756, −4.03733625596201950452762593564, −3.24846456485664692049307529787, −1.17907657034727866547869966177, −0.05013316636526785009384671227, 1.54395443229671068155392854587, 2.97000241598091434918591462942, 3.97016670693580164952058162786, 5.69205865635913258224922082558, 6.33856521516534093536668884772, 7.28120719911323368523250781457, 8.361391137454974389208944801400, 9.288169492964147163377259464144, 10.48580773636378245906762320883, 11.19504961861105747030690511582

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