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 − 1.43·5-s − 1.87·7-s + 8-s − 1.43·10-s − 5.50·11-s + 5.81·13-s − 1.87·14-s + 16-s + 3.16·17-s − 0.890·19-s − 1.43·20-s − 5.50·22-s − 1.68·23-s − 2.94·25-s + 5.81·26-s − 1.87·28-s + 6.77·29-s + 2.18·31-s + 32-s + 3.16·34-s + 2.68·35-s + 9.53·37-s − 0.890·38-s − 1.43·40-s − 5.50·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.641·5-s − 0.707·7-s + 0.353·8-s − 0.453·10-s − 1.66·11-s + 1.61·13-s − 0.500·14-s + 0.250·16-s + 0.766·17-s − 0.204·19-s − 0.320·20-s − 1.17·22-s − 0.351·23-s − 0.588·25-s + 1.13·26-s − 0.353·28-s + 1.25·29-s + 0.392·31-s + 0.176·32-s + 0.542·34-s + 0.453·35-s + 1.56·37-s − 0.144·38-s − 0.226·40-s − 0.859·41-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 \( 1 + 1.43T + 5T^{2} \)
7 \( 1 + 1.87T + 7T^{2} \)
11 \( 1 + 5.50T + 11T^{2} \)
13 \( 1 - 5.81T + 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 + 0.890T + 19T^{2} \)
23 \( 1 + 1.68T + 23T^{2} \)
29 \( 1 - 6.77T + 29T^{2} \)
31 \( 1 - 2.18T + 31T^{2} \)
37 \( 1 - 9.53T + 37T^{2} \)
41 \( 1 + 5.50T + 41T^{2} \)
43 \( 1 + 9.08T + 43T^{2} \)
47 \( 1 + 4.43T + 47T^{2} \)
53 \( 1 + 9.50T + 53T^{2} \)
59 \( 1 + 14.8T + 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 + 0.993T + 67T^{2} \)
71 \( 1 + 4.42T + 71T^{2} \)
73 \( 1 - 2.32T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + 15.1T + 83T^{2} \)
89 \( 1 - 3.41T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−8.071750791427634545464469122785, −7.40503633285576864599439032199, −6.26238013951809109167059377201, −6.04795519084846805760991546045, −4.97767703099070636403481815750, −4.28668423931572391692316515763, −3.23998008874760753875164178109, −2.97196068572264341363672821454, −1.51219697667423181271456436689, 0, 1.51219697667423181271456436689, 2.97196068572264341363672821454, 3.23998008874760753875164178109, 4.28668423931572391692316515763, 4.97767703099070636403481815750, 6.04795519084846805760991546045, 6.26238013951809109167059377201, 7.40503633285576864599439032199, 8.071750791427634545464469122785

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