Properties

Degree 2
Conductor $ 2 \cdot 13691 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s − 3·5-s + 2·6-s − 2·7-s − 8-s + 9-s + 3·10-s − 6·11-s − 2·12-s − 6·13-s + 2·14-s + 6·15-s + 16-s − 8·17-s − 18-s − 5·19-s − 3·20-s + 4·21-s + 6·22-s + 2·24-s + 4·25-s + 6·26-s + 4·27-s − 2·28-s + 2·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 1.80·11-s − 0.577·12-s − 1.66·13-s + 0.534·14-s + 1.54·15-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 1.14·19-s − 0.670·20-s + 0.872·21-s + 1.27·22-s + 0.408·24-s + 4/5·25-s + 1.17·26-s + 0.769·27-s − 0.377·28-s + 0.371·29-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 27382 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 27382 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(27382\)    =    \(2 \cdot 13691\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{27382} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 27382,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;13691\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;13691\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
13691 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.87582347637714, −15.70780124686833, −15.35614079179596, −14.81892398534507, −13.82355012214809, −13.11162112284388, −12.58867974955580, −12.28488832062087, −11.62418411877250, −11.20634735858314, −10.68914615363681, −10.19453953236317, −9.802370203255513, −8.815475997565422, −8.275011310751548, −7.923815118233866, −7.047577191738038, −6.755521562785137, −6.240244888362818, −5.247559203346068, −4.745270607684209, −4.380379024877120, −3.076726845990189, −2.765774972069143, −1.815865570564512, 0, 0, 0, 1.815865570564512, 2.765774972069143, 3.076726845990189, 4.380379024877120, 4.745270607684209, 5.247559203346068, 6.240244888362818, 6.755521562785137, 7.047577191738038, 7.923815118233866, 8.275011310751548, 8.815475997565422, 9.802370203255513, 10.19453953236317, 10.68914615363681, 11.20634735858314, 11.62418411877250, 12.28488832062087, 12.58867974955580, 13.11162112284388, 13.82355012214809, 14.81892398534507, 15.35614079179596, 15.70780124686833, 15.87582347637714

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