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Properties

Label 2-336-7.2-c5-0-38
Degree $2$
Conductor $336$
Sign $-0.636 - 0.770i$
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 + (46.4 − 80.3i)5-s + (118. + 52.8i)7-s + (−40.5 + 70.1i)9-s + (70.3 + 121. i)11-s − 1.11e3·13-s − 835.·15-s + (−27.4 − 47.5i)17-s + (−855. + 1.48e3i)19-s + (−120. − 1.16e3i)21-s + (−1.64e3 + 2.84e3i)23-s + (−2.74e3 − 4.75e3i)25-s + 729·27-s − 3.79e3·29-s + (−2.42e3 − 4.19e3i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.830 − 1.43i)5-s + (0.913 + 0.407i)7-s + (−0.166 + 0.288i)9-s + (0.175 + 0.303i)11-s − 1.82·13-s − 0.958·15-s + (−0.0230 − 0.0398i)17-s + (−0.543 + 0.942i)19-s + (−0.0596 − 0.574i)21-s + (−0.647 + 1.12i)23-s + (−0.878 − 1.52i)25-s + 0.192·27-s − 0.837·29-s + (−0.452 − 0.784i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.770i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.636 - 0.770i$
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: \((2,\ 336,\ (\ :5/2),\ -0.636 - 0.770i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.06142637436\)
\(L(\frac12)\) \(\approx\) \(0.06142637436\)
\(L(\frac{7}{2})\) 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 \( 1 + (-118. - 52.8i)T \)
good5 \( 1 + (-46.4 + 80.3i)T + (-1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (-70.3 - 121. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 1.11e3T + 3.71e5T^{2} \)
17 \( 1 + (27.4 + 47.5i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (855. - 1.48e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.64e3 - 2.84e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 3.79e3T + 2.05e7T^{2} \)
31 \( 1 + (2.42e3 + 4.19e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-5.68e3 + 9.84e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.03e4T + 1.15e8T^{2} \)
43 \( 1 + 7.13e3T + 1.47e8T^{2} \)
47 \( 1 + (8.20e3 - 1.42e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.04e4 - 1.81e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (1.81e4 + 3.13e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.47e3 - 4.28e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (1.14e4 + 1.98e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 2.63e4T + 1.80e9T^{2} \)
73 \( 1 + (2.76e4 + 4.79e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (2.49e4 - 4.32e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 4.48e4T + 3.93e9T^{2} \)
89 \( 1 + (6.39e4 - 1.10e5i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 6.56e4T + 8.58e9T^{2} \)
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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

−9.855270684184409818491703702198, −9.300763875490457942341269853771, −8.162525466857872147867132814399, −7.44485093541690016247182026617, −5.88435500701600825074985795204, −5.26042140031236965390038230975, −4.38108082617050744479993911256, −2.12942605284356581710850049282, −1.55333705832733998341213493784, −0.01463244889711078318694673993, 1.97572280678719246655680172426, 2.95550340944496905037957668999, 4.43663134930933180368147035898, 5.38855921398125688264115489478, 6.61994060754754164127203376606, 7.25169792644844305121860135776, 8.570667174237721698255070944382, 9.884671350567653325769207689171, 10.30345277014334884895268810255, 11.15488618618743467931141094734

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