Properties

Degree 2
Conductor $ 2 \cdot 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  − 2-s − 3-s + 4-s + 1.72·5-s + 6-s + 2.39·7-s − 8-s + 9-s − 1.72·10-s − 4.25·11-s − 12-s + 13-s − 2.39·14-s − 1.72·15-s + 16-s + 6.64·17-s − 18-s + 1.39·19-s + 1.72·20-s − 2.39·21-s + 4.25·22-s + 0.660·23-s + 24-s − 2.03·25-s − 26-s − 27-s + 2.39·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.769·5-s + 0.408·6-s + 0.903·7-s − 0.353·8-s + 0.333·9-s − 0.544·10-s − 1.28·11-s − 0.288·12-s + 0.277·13-s − 0.638·14-s − 0.444·15-s + 0.250·16-s + 1.61·17-s − 0.235·18-s + 0.318·19-s + 0.384·20-s − 0.521·21-s + 0.906·22-s + 0.137·23-s + 0.204·24-s − 0.407·25-s − 0.196·26-s − 0.192·27-s + 0.451·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8034\)    =    \(2 \cdot 3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8034,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;13,\;103\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13,\;103\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + T \)
3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 - T \)
good5 \( 1 - 1.72T + 5T^{2} \)
7 \( 1 - 2.39T + 7T^{2} \)
11 \( 1 + 4.25T + 11T^{2} \)
17 \( 1 - 6.64T + 17T^{2} \)
19 \( 1 - 1.39T + 19T^{2} \)
23 \( 1 - 0.660T + 23T^{2} \)
29 \( 1 + 7.21T + 29T^{2} \)
31 \( 1 + 7.06T + 31T^{2} \)
37 \( 1 - 0.542T + 37T^{2} \)
41 \( 1 - 6.29T + 41T^{2} \)
43 \( 1 + 12.5T + 43T^{2} \)
47 \( 1 + 5.06T + 47T^{2} \)
53 \( 1 - 3.16T + 53T^{2} \)
59 \( 1 - 0.644T + 59T^{2} \)
61 \( 1 - 7.01T + 61T^{2} \)
67 \( 1 + 8.17T + 67T^{2} \)
71 \( 1 - 1.21T + 71T^{2} \)
73 \( 1 + 1.21T + 73T^{2} \)
79 \( 1 + 5.16T + 79T^{2} \)
83 \( 1 + 3.97T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−7.66667656221158738613175457345, −6.95251978835286772283597707778, −5.91273149018515663826851653407, −5.46678202843149369370223826114, −5.08177966246653373125682033532, −3.84697083873084914402180276043, −2.91116779127382566635657858161, −1.88550776225120894918549782352, −1.32894695600967012567916634250, 0, 1.32894695600967012567916634250, 1.88550776225120894918549782352, 2.91116779127382566635657858161, 3.84697083873084914402180276043, 5.08177966246653373125682033532, 5.46678202843149369370223826114, 5.91273149018515663826851653407, 6.95251978835286772283597707778, 7.66667656221158738613175457345

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