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Properties

Label 2-336-12.11-c5-0-19
Degree $2$
Conductor $336$
Sign $-0.877 - 0.478i$
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  + (−7.46 + 13.6i)3-s + 31.0i·5-s + 49i·7-s + (−131. − 204. i)9-s + 273.·11-s + 431.·13-s + (−424. − 231. i)15-s + 648. i·17-s + 2.30e3i·19-s + (−670. − 365. i)21-s + 3.72e3·23-s + 2.16e3·25-s + (3.77e3 − 274. i)27-s + 7.00e3i·29-s − 1.72e3i·31-s + ⋯
L(s)  = 1  + (−0.478 + 0.877i)3-s + 0.554i·5-s + 0.377i·7-s + (−0.541 − 0.840i)9-s + 0.680·11-s + 0.708·13-s + (−0.486 − 0.265i)15-s + 0.544i·17-s + 1.46i·19-s + (−0.331 − 0.181i)21-s + 1.46·23-s + 0.692·25-s + (0.997 − 0.0723i)27-s + 1.54i·29-s − 0.322i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.478i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.877 - 0.478i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ -0.877 - 0.478i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.599892995\)
\(L(\frac12)\) \(\approx\) \(1.599892995\)
\(L(\frac{7}{2})\) 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 \( 1 + (7.46 - 13.6i)T \)
7 \( 1 - 49iT \)
good5 \( 1 - 31.0iT - 3.12e3T^{2} \)
11 \( 1 - 273.T + 1.61e5T^{2} \)
13 \( 1 - 431.T + 3.71e5T^{2} \)
17 \( 1 - 648. iT - 1.41e6T^{2} \)
19 \( 1 - 2.30e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.72e3T + 6.43e6T^{2} \)
29 \( 1 - 7.00e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.72e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.22e4T + 6.93e7T^{2} \)
41 \( 1 + 1.66e4iT - 1.15e8T^{2} \)
43 \( 1 - 9.18e3iT - 1.47e8T^{2} \)
47 \( 1 - 2.32e4T + 2.29e8T^{2} \)
53 \( 1 - 1.91e4iT - 4.18e8T^{2} \)
59 \( 1 - 1.03e4T + 7.14e8T^{2} \)
61 \( 1 + 7.05e3T + 8.44e8T^{2} \)
67 \( 1 + 4.61e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.59e4T + 1.80e9T^{2} \)
73 \( 1 + 5.11e4T + 2.07e9T^{2} \)
79 \( 1 - 1.06e4iT - 3.07e9T^{2} \)
83 \( 1 - 6.75e4T + 3.93e9T^{2} \)
89 \( 1 - 7.33e4iT - 5.58e9T^{2} \)
97 \( 1 + 1.34e5T + 8.58e9T^{2} \)
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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

−10.73486720385611442455124385419, −10.58210909759988946993360758069, −9.190989470223700092415579688826, −8.649397589502309823894958726179, −7.09357917006337208741881973573, −6.14564600161974753559646952875, −5.27846358951135628295567176978, −3.95739494579378857883922664173, −3.13325638942362249827475710812, −1.31613767352388530754515189236, 0.51336832573608185852432090710, 1.31483080086695799086820037546, 2.84208721762975740231967984598, 4.45371700839426727125233820058, 5.41524465824240818456266116270, 6.63835599994528283945214646074, 7.22327280798519055389670522057, 8.502476597024468503012514956317, 9.166075354273378441828158262232, 10.57934306398198466200771466755

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