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  + 0.414·2-s − 3-s − 1.82·4-s − 0.585·5-s − 0.414·6-s − 2.82·7-s − 1.58·8-s + 9-s − 0.242·10-s − 1.41·11-s + 1.82·12-s + 13-s − 1.17·14-s + 0.585·15-s + 3·16-s + 6.82·17-s + 0.414·18-s − 5.65·19-s + 1.07·20-s + 2.82·21-s − 0.585·22-s + 7.65·23-s + 1.58·24-s − 4.65·25-s + 0.414·26-s − 27-s + 5.17·28-s + ⋯
L(s)  = 1  + 0.292·2-s − 0.577·3-s − 0.914·4-s − 0.261·5-s − 0.169·6-s − 1.06·7-s − 0.560·8-s + 0.333·9-s − 0.0767·10-s − 0.426·11-s + 0.527·12-s + 0.277·13-s − 0.313·14-s + 0.151·15-s + 0.750·16-s + 1.65·17-s + 0.0976·18-s − 1.29·19-s + 0.239·20-s + 0.617·21-s − 0.124·22-s + 1.59·23-s + 0.323·24-s − 0.931·25-s + 0.0812·26-s − 0.192·27-s + 0.977·28-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 \( 1 - 0.414T + 2T^{2} \)
5 \( 1 + 0.585T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 + 5.65T + 19T^{2} \)
23 \( 1 - 7.65T + 23T^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 - 1.17T + 41T^{2} \)
43 \( 1 + 2.82T + 43T^{2} \)
47 \( 1 - 7.07T + 47T^{2} \)
53 \( 1 + 3.65T + 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 4.58T + 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + 7.75T + 73T^{2} \)
79 \( 1 + 6.48T + 79T^{2} \)
83 \( 1 + 0.828T + 83T^{2} \)
89 \( 1 - 5.75T + 89T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
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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

−8.038081041354474494667567838678, −7.37097537548947672123417905738, −6.34065749901947973446936871391, −5.85570078586470964474943916454, −5.10088895757236015877013821373, −4.27849445892947347507671758975, −3.54047749194034220661167476773, −2.78290586070126621428328065555, −1.06300995681247488470159513719, 0, 1.06300995681247488470159513719, 2.78290586070126621428328065555, 3.54047749194034220661167476773, 4.27849445892947347507671758975, 5.10088895757236015877013821373, 5.85570078586470964474943916454, 6.34065749901947973446936871391, 7.37097537548947672123417905738, 8.038081041354474494667567838678

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