Properties

Degree 2
Conductor $ 3 \cdot 13 \cdot 103 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·4-s + 5-s + 2·7-s + 9-s − 2·12-s − 13-s + 15-s + 4·16-s − 3·17-s − 4·19-s − 2·20-s + 2·21-s − 8·23-s − 4·25-s + 27-s − 4·28-s + 5·29-s − 31-s + 2·35-s − 2·36-s + 37-s − 39-s − 6·41-s − 8·43-s + 45-s + 3·47-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.577·12-s − 0.277·13-s + 0.258·15-s + 16-s − 0.727·17-s − 0.917·19-s − 0.447·20-s + 0.436·21-s − 1.66·23-s − 4/5·25-s + 0.192·27-s − 0.755·28-s + 0.928·29-s − 0.179·31-s + 0.338·35-s − 1/3·36-s + 0.164·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4017\)    =    \(3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4017} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4017,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;13,\;103\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;13,\;103\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 - T \)
13 \( 1 + T \)
103 \( 1 - T \)
good2 \( 1 + p T^{2} \)
5 \( 1 - T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 - T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−8.289812019619759109423310079486, −7.62363466505198381248563064317, −6.55774121903757458544896035936, −5.80653174799998452773628812102, −4.86520020905111913093774715807, −4.34415681869639972823357680004, −3.55991169571847410774924610306, −2.33128840435454651393386049365, −1.59002493368593219071271688421, 0, 1.59002493368593219071271688421, 2.33128840435454651393386049365, 3.55991169571847410774924610306, 4.34415681869639972823357680004, 4.86520020905111913093774715807, 5.80653174799998452773628812102, 6.55774121903757458544896035936, 7.62363466505198381248563064317, 8.289812019619759109423310079486

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