Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 223 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 1.64·5-s − 3.78·7-s + 8-s − 1.64·10-s + 0.948·11-s + 2.09·13-s − 3.78·14-s + 16-s + 0.0270·17-s + 5.38·19-s − 1.64·20-s + 0.948·22-s + 3.61·23-s − 2.30·25-s + 2.09·26-s − 3.78·28-s − 3.17·29-s − 7.38·31-s + 32-s + 0.0270·34-s + 6.21·35-s − 2.49·37-s + 5.38·38-s − 1.64·40-s + 4.22·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.733·5-s − 1.43·7-s + 0.353·8-s − 0.518·10-s + 0.285·11-s + 0.580·13-s − 1.01·14-s + 0.250·16-s + 0.00656·17-s + 1.23·19-s − 0.366·20-s + 0.202·22-s + 0.754·23-s − 0.461·25-s + 0.410·26-s − 0.715·28-s − 0.589·29-s − 1.32·31-s + 0.176·32-s + 0.00464·34-s + 1.05·35-s − 0.410·37-s + 0.873·38-s − 0.259·40-s + 0.659·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4014\)    =    \(2 \cdot 3^{2} \cdot 223\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4014} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4014,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;223\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;223\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 \)
223 \( 1 + T \)
good5 \( 1 + 1.64T + 5T^{2} \)
7 \( 1 + 3.78T + 7T^{2} \)
11 \( 1 - 0.948T + 11T^{2} \)
13 \( 1 - 2.09T + 13T^{2} \)
17 \( 1 - 0.0270T + 17T^{2} \)
19 \( 1 - 5.38T + 19T^{2} \)
23 \( 1 - 3.61T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 + 7.38T + 31T^{2} \)
37 \( 1 + 2.49T + 37T^{2} \)
41 \( 1 - 4.22T + 41T^{2} \)
43 \( 1 + 6.30T + 43T^{2} \)
47 \( 1 + 2.59T + 47T^{2} \)
53 \( 1 + 5.09T + 53T^{2} \)
59 \( 1 + 9.78T + 59T^{2} \)
61 \( 1 + 2.50T + 61T^{2} \)
67 \( 1 + 4.35T + 67T^{2} \)
71 \( 1 - 8.97T + 71T^{2} \)
73 \( 1 + 11.0T + 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 - 8.87T + 83T^{2} \)
89 \( 1 - 0.793T + 89T^{2} \)
97 \( 1 + 8.13T + 97T^{2} \)
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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

−7.81942638425135464907906853534, −7.26256044451630801077329446622, −6.55612770835210228108912160299, −5.86821876256841211923549038090, −5.10882068020397855925354803669, −4.02950398793090092217253431249, −3.46452530421827535140746085749, −2.92678435209176391308625275471, −1.47426509677664598195120476195, 0, 1.47426509677664598195120476195, 2.92678435209176391308625275471, 3.46452530421827535140746085749, 4.02950398793090092217253431249, 5.10882068020397855925354803669, 5.86821876256841211923549038090, 6.55612770835210228108912160299, 7.26256044451630801077329446622, 7.81942638425135464907906853534

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