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Properties

Label 2-336-7.2-c5-0-13
Degree $2$
Conductor $336$
Sign $0.972 - 0.233i$
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 + (37.7 − 65.3i)5-s + (−99.4 + 83.1i)7-s + (−40.5 + 70.1i)9-s + (−74.7 − 129. i)11-s + 349.·13-s − 679.·15-s + (574. + 995. i)17-s + (−1.39e3 + 2.42e3i)19-s + (1.09e3 + 401. i)21-s + (906. − 1.57e3i)23-s + (−1.28e3 − 2.22e3i)25-s + 729·27-s − 759.·29-s + (4.51e3 + 7.82e3i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.675 − 1.16i)5-s + (−0.767 + 0.641i)7-s + (−0.166 + 0.288i)9-s + (−0.186 − 0.322i)11-s + 0.573·13-s − 0.779·15-s + (0.482 + 0.835i)17-s + (−0.888 + 1.53i)19-s + (0.542 + 0.198i)21-s + (0.357 − 0.619i)23-s + (−0.411 − 0.713i)25-s + 0.192·27-s − 0.167·29-s + (0.843 + 1.46i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.972 - 0.233i$
Analytic conductor: \(53.8889\)
Root analytic conductor: \(7.34091\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :5/2),\ 0.972 - 0.233i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.626331584\)
\(L(\frac12)\) \(\approx\) \(1.626331584\)
\(L(\frac{7}{2})\) 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 \( 1 + (99.4 - 83.1i)T \)
good5 \( 1 + (-37.7 + 65.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (74.7 + 129. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 349.T + 3.71e5T^{2} \)
17 \( 1 + (-574. - 995. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.39e3 - 2.42e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (-906. + 1.57e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 759.T + 2.05e7T^{2} \)
31 \( 1 + (-4.51e3 - 7.82e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (3.89e3 - 6.75e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 - 7.64e3T + 1.15e8T^{2} \)
43 \( 1 + 1.21e4T + 1.47e8T^{2} \)
47 \( 1 + (-1.22e4 + 2.13e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (6.79e3 + 1.17e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.31e4 + 2.28e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.76e4 - 3.05e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.71e4 - 4.70e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 7.01e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.22e4 - 3.85e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-3.08e4 + 5.33e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 8.71e4T + 3.93e9T^{2} \)
89 \( 1 + (4.92e4 - 8.53e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 3.23e4T + 8.58e9T^{2} \)
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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

−10.62218332178224995604072852903, −9.855684380858355099784442124078, −8.619437599632245049616289137901, −8.328901825825577667572612221965, −6.58710765598126271228871293941, −5.90946677805727893461049424890, −5.08141941541357539426494216769, −3.55264311809499037253193918691, −2.00646339001160904213629184629, −0.974320334314310470347957917689, 0.52552397027707958956902202247, 2.45068996262327408558974267907, 3.39954939959838367557642014922, 4.64945514098815285281246171749, 6.00202521184284680332592697544, 6.69144271671169524909719445782, 7.58876696990920585690303417496, 9.265114367739517182586482014100, 9.769250592106327629818084999368, 10.80725770303247811433520051120

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