Properties

Degree 2
Conductor $ 2 \cdot 7 \cdot 13^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.24·3-s + 4-s + 1.69·5-s + 2.24·6-s − 7-s + 8-s + 2.04·9-s + 1.69·10-s + 0.445·11-s + 2.24·12-s − 14-s + 3.80·15-s + 16-s − 2.15·17-s + 2.04·18-s + 6.35·19-s + 1.69·20-s − 2.24·21-s + 0.445·22-s − 0.911·23-s + 2.24·24-s − 2.13·25-s − 2.13·27-s − 28-s + 3.58·29-s + 3.80·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.756·5-s + 0.917·6-s − 0.377·7-s + 0.353·8-s + 0.682·9-s + 0.535·10-s + 0.134·11-s + 0.648·12-s − 0.267·14-s + 0.981·15-s + 0.250·16-s − 0.523·17-s + 0.482·18-s + 1.45·19-s + 0.378·20-s − 0.490·21-s + 0.0948·22-s − 0.190·23-s + 0.458·24-s − 0.427·25-s − 0.411·27-s − 0.188·28-s + 0.665·29-s + 0.694·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2366\)    =    \(2 \cdot 7 \cdot 13^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{2366} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2366,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(4.991135146\)
\(L(\frac12)\)  \(\approx\)  \(4.991135146\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;13\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;13\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 + T \)
13 \( 1 \)
good3 \( 1 - 2.24T + 3T^{2} \)
5 \( 1 - 1.69T + 5T^{2} \)
11 \( 1 - 0.445T + 11T^{2} \)
17 \( 1 + 2.15T + 17T^{2} \)
19 \( 1 - 6.35T + 19T^{2} \)
23 \( 1 + 0.911T + 23T^{2} \)
29 \( 1 - 3.58T + 29T^{2} \)
31 \( 1 - 8.89T + 31T^{2} \)
37 \( 1 - 10.8T + 37T^{2} \)
41 \( 1 - 2.41T + 41T^{2} \)
43 \( 1 - 4.63T + 43T^{2} \)
47 \( 1 + 9.75T + 47T^{2} \)
53 \( 1 + 8.74T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 - 1.37T + 61T^{2} \)
67 \( 1 - 6.23T + 67T^{2} \)
71 \( 1 + 5.76T + 71T^{2} \)
73 \( 1 + 9.93T + 73T^{2} \)
79 \( 1 + 6.30T + 79T^{2} \)
83 \( 1 + 2.91T + 83T^{2} \)
89 \( 1 - 18.1T + 89T^{2} \)
97 \( 1 + 3.30T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.111037609855064053661027352030, −8.081853903830939703635402939416, −7.58677532408271765610362148939, −6.48511654029689579116965473402, −5.95982217573503243218932671641, −4.88145457634050711964668802649, −4.02012316849193116834220697501, −2.99465980874160581162293047811, −2.58060668054311888099501511624, −1.40850576997079563377082833895, 1.40850576997079563377082833895, 2.58060668054311888099501511624, 2.99465980874160581162293047811, 4.02012316849193116834220697501, 4.88145457634050711964668802649, 5.95982217573503243218932671641, 6.48511654029689579116965473402, 7.58677532408271765610362148939, 8.081853903830939703635402939416, 9.111037609855064053661027352030

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