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 − 1.54·5-s − 1.28·7-s + 8-s − 1.54·10-s + 2.49·11-s − 2.65·13-s − 1.28·14-s + 16-s − 1.92·17-s + 4.69·19-s − 1.54·20-s + 2.49·22-s − 3.63·23-s − 2.61·25-s − 2.65·26-s − 1.28·28-s − 9.08·29-s + 5.96·31-s + 32-s − 1.92·34-s + 1.99·35-s + 5.70·37-s + 4.69·38-s − 1.54·40-s − 6.53·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 0.690·5-s − 0.487·7-s + 0.353·8-s − 0.488·10-s + 0.752·11-s − 0.737·13-s − 0.344·14-s + 0.250·16-s − 0.465·17-s + 1.07·19-s − 0.345·20-s + 0.531·22-s − 0.757·23-s − 0.522·25-s − 0.521·26-s − 0.243·28-s − 1.68·29-s + 1.07·31-s + 0.176·32-s − 0.329·34-s + 0.336·35-s + 0.938·37-s + 0.761·38-s − 0.244·40-s − 1.02·41-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 \( 1 + 1.54T + 5T^{2} \)
7 \( 1 + 1.28T + 7T^{2} \)
11 \( 1 - 2.49T + 11T^{2} \)
13 \( 1 + 2.65T + 13T^{2} \)
17 \( 1 + 1.92T + 17T^{2} \)
19 \( 1 - 4.69T + 19T^{2} \)
23 \( 1 + 3.63T + 23T^{2} \)
29 \( 1 + 9.08T + 29T^{2} \)
31 \( 1 - 5.96T + 31T^{2} \)
37 \( 1 - 5.70T + 37T^{2} \)
41 \( 1 + 6.53T + 41T^{2} \)
43 \( 1 - 6.44T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + 2.97T + 53T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 - 4.51T + 67T^{2} \)
71 \( 1 + 6.01T + 71T^{2} \)
73 \( 1 + 9.46T + 73T^{2} \)
79 \( 1 + 9.23T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 - 0.168T + 89T^{2} \)
97 \( 1 - 1.05T + 97T^{2} \)
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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

−7.81104655377502901846153790922, −7.36408232336366704704691301389, −6.52648080594366956587129907549, −5.88742138470388163245799491340, −4.96604493781432318476151352138, −4.18310861404999809440126400088, −3.55818512553463386015267206431, −2.71371680099437928475709910910, −1.56700354287063044159040422494, 0, 1.56700354287063044159040422494, 2.71371680099437928475709910910, 3.55818512553463386015267206431, 4.18310861404999809440126400088, 4.96604493781432318476151352138, 5.88742138470388163245799491340, 6.52648080594366956587129907549, 7.36408232336366704704691301389, 7.81104655377502901846153790922

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