Properties

Degree 2
Conductor $ 3 \cdot 13 \cdot 103 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.27·2-s − 3-s − 0.372·4-s − 0.506·5-s + 1.27·6-s − 3.05·7-s + 3.02·8-s + 9-s + 0.645·10-s − 1.77·11-s + 0.372·12-s + 13-s + 3.89·14-s + 0.506·15-s − 3.11·16-s − 3.59·17-s − 1.27·18-s − 0.715·19-s + 0.188·20-s + 3.05·21-s + 2.26·22-s − 0.238·23-s − 3.02·24-s − 4.74·25-s − 1.27·26-s − 27-s + 1.13·28-s + ⋯
L(s)  = 1  − 0.902·2-s − 0.577·3-s − 0.186·4-s − 0.226·5-s + 0.520·6-s − 1.15·7-s + 1.07·8-s + 0.333·9-s + 0.204·10-s − 0.535·11-s + 0.107·12-s + 0.277·13-s + 1.04·14-s + 0.130·15-s − 0.779·16-s − 0.871·17-s − 0.300·18-s − 0.164·19-s + 0.0421·20-s + 0.666·21-s + 0.483·22-s − 0.0497·23-s − 0.617·24-s − 0.948·25-s − 0.250·26-s − 0.192·27-s + 0.214·28-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  =  0
Selberg data  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2288658312$
$L(\frac12)$  $\approx$  $0.2288658312$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 + T \)
good2 \( 1 + 1.27T + 2T^{2} \)
5 \( 1 + 0.506T + 5T^{2} \)
7 \( 1 + 3.05T + 7T^{2} \)
11 \( 1 + 1.77T + 11T^{2} \)
17 \( 1 + 3.59T + 17T^{2} \)
19 \( 1 + 0.715T + 19T^{2} \)
23 \( 1 + 0.238T + 23T^{2} \)
29 \( 1 - 2.95T + 29T^{2} \)
31 \( 1 - 0.136T + 31T^{2} \)
37 \( 1 - 5.90T + 37T^{2} \)
41 \( 1 + 5.81T + 41T^{2} \)
43 \( 1 + 2.36T + 43T^{2} \)
47 \( 1 + 7.27T + 47T^{2} \)
53 \( 1 + 4.12T + 53T^{2} \)
59 \( 1 + 4.06T + 59T^{2} \)
61 \( 1 + 3.05T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 - 6.06T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 - 1.65T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−8.406688426484965562826417906441, −7.910875312910079008056334614521, −6.95590392802035352787287590824, −6.44743406373552587319281208290, −5.56067504498333197314954569273, −4.64122085390894371817764036366, −3.94323090499832305255747826075, −2.88669602465494498735075058343, −1.64556809164748647723040584888, −0.32392210636250452958421956677, 0.32392210636250452958421956677, 1.64556809164748647723040584888, 2.88669602465494498735075058343, 3.94323090499832305255747826075, 4.64122085390894371817764036366, 5.56067504498333197314954569273, 6.44743406373552587319281208290, 6.95590392802035352787287590824, 7.910875312910079008056334614521, 8.406688426484965562826417906441

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