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 − 0.686·5-s + 2.87·7-s + 8-s − 0.686·10-s − 5.51·11-s + 0.329·13-s + 2.87·14-s + 16-s − 4.44·17-s − 1.53·19-s − 0.686·20-s − 5.51·22-s + 3.17·23-s − 4.52·25-s + 0.329·26-s + 2.87·28-s − 7.03·29-s − 7.43·31-s + 32-s − 4.44·34-s − 1.97·35-s − 7.58·37-s − 1.53·38-s − 0.686·40-s + 2.51·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.306·5-s + 1.08·7-s + 0.353·8-s − 0.217·10-s − 1.66·11-s + 0.0914·13-s + 0.768·14-s + 0.250·16-s − 1.07·17-s − 0.352·19-s − 0.153·20-s − 1.17·22-s + 0.661·23-s − 0.905·25-s + 0.0646·26-s + 0.543·28-s − 1.30·29-s − 1.33·31-s + 0.176·32-s − 0.761·34-s − 0.333·35-s − 1.24·37-s − 0.249·38-s − 0.108·40-s + 0.393·41-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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
223 \( 1 + T \)
good5 \( 1 + 0.686T + 5T^{2} \)
7 \( 1 - 2.87T + 7T^{2} \)
11 \( 1 + 5.51T + 11T^{2} \)
13 \( 1 - 0.329T + 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 + 1.53T + 19T^{2} \)
23 \( 1 - 3.17T + 23T^{2} \)
29 \( 1 + 7.03T + 29T^{2} \)
31 \( 1 + 7.43T + 31T^{2} \)
37 \( 1 + 7.58T + 37T^{2} \)
41 \( 1 - 2.51T + 41T^{2} \)
43 \( 1 - 6.97T + 43T^{2} \)
47 \( 1 + 6.00T + 47T^{2} \)
53 \( 1 - 0.840T + 53T^{2} \)
59 \( 1 - 3.14T + 59T^{2} \)
61 \( 1 + 2.68T + 61T^{2} \)
67 \( 1 + 13.9T + 67T^{2} \)
71 \( 1 - 6.67T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 5.22T + 83T^{2} \)
89 \( 1 + 8.29T + 89T^{2} \)
97 \( 1 - 4.45T + 97T^{2} \)
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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

−7.82046288614393937529062494565, −7.52147501100821437939762238140, −6.58918930230629009373108453835, −5.51076514739119211144986029124, −5.16203934471662344695036348283, −4.34232724473818014368158973394, −3.54729462559134042702247929972, −2.43329179631741334718209031751, −1.78239988901328000640044416830, 0, 1.78239988901328000640044416830, 2.43329179631741334718209031751, 3.54729462559134042702247929972, 4.34232724473818014368158973394, 5.16203934471662344695036348283, 5.51076514739119211144986029124, 6.58918930230629009373108453835, 7.52147501100821437939762238140, 7.82046288614393937529062494565

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