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Properties

Label 2-336-7.6-c8-0-7
Degree $2$
Conductor $336$
Sign $-0.725 + 0.688i$
Analytic cond. $136.879$
Root an. cond. $11.6995$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 46.7i·3-s + 747. i·5-s + (1.65e3 + 1.74e3i)7-s − 2.18e3·9-s + 2.18e4·11-s − 2.84e4i·13-s − 3.49e4·15-s − 5.73e4i·17-s − 3.05e4i·19-s + (−8.14e4 + 7.72e4i)21-s − 4.81e5·23-s − 1.67e5·25-s − 1.02e5i·27-s − 6.06e4·29-s + 6.69e5i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.19i·5-s + (0.688 + 0.725i)7-s − 0.333·9-s + 1.49·11-s − 0.995i·13-s − 0.690·15-s − 0.686i·17-s − 0.234i·19-s + (−0.418 + 0.397i)21-s − 1.72·23-s − 0.429·25-s − 0.192i·27-s − 0.0857·29-s + 0.724i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.725 + 0.688i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.725 + 0.688i$
Analytic conductor: \(136.879\)
Root analytic conductor: \(11.6995\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :4),\ -0.725 + 0.688i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.7246105644\)
\(L(\frac12)\) \(\approx\) \(0.7246105644\)
\(L(5)\) 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 \( 1 - 46.7iT \)
7 \( 1 + (-1.65e3 - 1.74e3i)T \)
good5 \( 1 - 747. iT - 3.90e5T^{2} \)
11 \( 1 - 2.18e4T + 2.14e8T^{2} \)
13 \( 1 + 2.84e4iT - 8.15e8T^{2} \)
17 \( 1 + 5.73e4iT - 6.97e9T^{2} \)
19 \( 1 + 3.05e4iT - 1.69e10T^{2} \)
23 \( 1 + 4.81e5T + 7.83e10T^{2} \)
29 \( 1 + 6.06e4T + 5.00e11T^{2} \)
31 \( 1 - 6.69e5iT - 8.52e11T^{2} \)
37 \( 1 + 2.73e6T + 3.51e12T^{2} \)
41 \( 1 - 5.07e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.66e6T + 1.16e13T^{2} \)
47 \( 1 + 8.63e6iT - 2.38e13T^{2} \)
53 \( 1 + 6.93e6T + 6.22e13T^{2} \)
59 \( 1 + 3.33e6iT - 1.46e14T^{2} \)
61 \( 1 - 3.40e6iT - 1.91e14T^{2} \)
67 \( 1 + 2.35e7T + 4.06e14T^{2} \)
71 \( 1 + 4.46e7T + 6.45e14T^{2} \)
73 \( 1 + 1.26e7iT - 8.06e14T^{2} \)
79 \( 1 - 5.19e6T + 1.51e15T^{2} \)
83 \( 1 + 4.65e6iT - 2.25e15T^{2} \)
89 \( 1 - 4.37e7iT - 3.93e15T^{2} \)
97 \( 1 - 1.46e8iT - 7.83e15T^{2} \)
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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

−10.69371796542494346161044728013, −9.939241745937548252715691892433, −8.946411390014265561602620968685, −8.021596017630797574028289688365, −6.85272342935589284175220685541, −5.97887638737062103437148466058, −4.89892228116323554145179805837, −3.66860907398535623672451285327, −2.80150446462187903108087916704, −1.57882858824161511855960542556, 0.13354086722533566279680324674, 1.41941942027765789485994427636, 1.72368735463585051406533953950, 3.89403219237762638921370335125, 4.43753739907513092570780084411, 5.77540309544075242699977532191, 6.74420703071183918743814877051, 7.79260199339056045869406391792, 8.650228776693285711241509152238, 9.351765097575279306282334082138

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