Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 223 $
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 + 4-s + 2.60·7-s + 8-s + 6.30·11-s + 5.30·13-s + 2.60·14-s + 16-s + 4.60·17-s + 0.697·19-s + 6.30·22-s − 4·23-s − 5·25-s + 5.30·26-s + 2.60·28-s + 1.69·29-s − 0.605·31-s + 32-s + 4.60·34-s − 2.60·37-s + 0.697·38-s − 4·41-s − 6.90·43-s + 6.30·44-s − 4·46-s + 4.30·47-s − 0.211·49-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.984·7-s + 0.353·8-s + 1.90·11-s + 1.47·13-s + 0.696·14-s + 0.250·16-s + 1.11·17-s + 0.159·19-s + 1.34·22-s − 0.834·23-s − 25-s + 1.03·26-s + 0.492·28-s + 0.315·29-s − 0.108·31-s + 0.176·32-s + 0.789·34-s − 0.428·37-s + 0.113·38-s − 0.624·41-s − 1.05·43-s + 0.950·44-s − 0.589·46-s + 0.627·47-s − 0.0301·49-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  =  0
Selberg data  =  $(2,\ 4014,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.338704375$
$L(\frac12)$  $\approx$  $4.338704375$
$L(\frac{3}{2})$   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 \( 1 + 5T^{2} \)
7 \( 1 - 2.60T + 7T^{2} \)
11 \( 1 - 6.30T + 11T^{2} \)
13 \( 1 - 5.30T + 13T^{2} \)
17 \( 1 - 4.60T + 17T^{2} \)
19 \( 1 - 0.697T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 1.69T + 29T^{2} \)
31 \( 1 + 0.605T + 31T^{2} \)
37 \( 1 + 2.60T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 6.90T + 43T^{2} \)
47 \( 1 - 4.30T + 47T^{2} \)
53 \( 1 + 3.90T + 53T^{2} \)
59 \( 1 + 3.30T + 59T^{2} \)
61 \( 1 + 6.30T + 61T^{2} \)
67 \( 1 + 2.60T + 67T^{2} \)
71 \( 1 + 3.21T + 71T^{2} \)
73 \( 1 - 1.90T + 73T^{2} \)
79 \( 1 + 9.51T + 79T^{2} \)
83 \( 1 - 9.21T + 83T^{2} \)
89 \( 1 - 2.78T + 89T^{2} \)
97 \( 1 + 15.2T + 97T^{2} \)
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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

−8.327948808020895528790761162709, −7.75829950159078180381952489958, −6.78108393961842253342077941767, −6.13742942172902619036742078109, −5.54958076003078191977393141470, −4.54032896007714357708221393900, −3.85513249980711232195938008799, −3.30996906844965300312688807378, −1.74274490168439221789530784475, −1.29875341973171179208727592951, 1.29875341973171179208727592951, 1.74274490168439221789530784475, 3.30996906844965300312688807378, 3.85513249980711232195938008799, 4.54032896007714357708221393900, 5.54958076003078191977393141470, 6.13742942172902619036742078109, 6.78108393961842253342077941767, 7.75829950159078180381952489958, 8.327948808020895528790761162709

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