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  − 2.65·2-s − 3-s + 5.03·4-s + 3.62·5-s + 2.65·6-s − 1.69·7-s − 8.06·8-s + 9-s − 9.62·10-s + 0.122·11-s − 5.03·12-s + 13-s + 4.50·14-s − 3.62·15-s + 11.3·16-s + 1.64·17-s − 2.65·18-s + 2.48·19-s + 18.2·20-s + 1.69·21-s − 0.325·22-s + 7.71·23-s + 8.06·24-s + 8.14·25-s − 2.65·26-s − 27-s − 8.55·28-s + ⋯
L(s)  = 1  − 1.87·2-s − 0.577·3-s + 2.51·4-s + 1.62·5-s + 1.08·6-s − 0.641·7-s − 2.85·8-s + 0.333·9-s − 3.04·10-s + 0.0369·11-s − 1.45·12-s + 0.277·13-s + 1.20·14-s − 0.936·15-s + 2.82·16-s + 0.399·17-s − 0.625·18-s + 0.570·19-s + 4.08·20-s + 0.370·21-s − 0.0693·22-s + 1.60·23-s + 1.64·24-s + 1.62·25-s − 0.520·26-s − 0.192·27-s − 1.61·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.8833075407$
$L(\frac12)$  $\approx$  $0.8833075407$
$L(\frac{3}{2})$   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 \( 1 + 2.65T + 2T^{2} \)
5 \( 1 - 3.62T + 5T^{2} \)
7 \( 1 + 1.69T + 7T^{2} \)
11 \( 1 - 0.122T + 11T^{2} \)
17 \( 1 - 1.64T + 17T^{2} \)
19 \( 1 - 2.48T + 19T^{2} \)
23 \( 1 - 7.71T + 23T^{2} \)
29 \( 1 + 8.37T + 29T^{2} \)
31 \( 1 + 5.04T + 31T^{2} \)
37 \( 1 - 7.37T + 37T^{2} \)
41 \( 1 + 5.29T + 41T^{2} \)
43 \( 1 - 5.82T + 43T^{2} \)
47 \( 1 - 11.9T + 47T^{2} \)
53 \( 1 + 3.27T + 53T^{2} \)
59 \( 1 - 9.14T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 + 6.39T + 73T^{2} \)
79 \( 1 - 9.16T + 79T^{2} \)
83 \( 1 - 16.7T + 83T^{2} \)
89 \( 1 - 1.34T + 89T^{2} \)
97 \( 1 - 8.30T + 97T^{2} \)
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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

−8.838945727981487872032771073719, −7.62050216992773679361166762160, −7.15582339612203780987701925291, −6.30579036571721507470204211305, −5.89986463181133977521305231434, −5.14564733171739913917542697349, −3.39786250515803182705520575721, −2.46703708838501200953500525247, −1.60121592086931748901383574825, −0.76985518137393737681112426385, 0.76985518137393737681112426385, 1.60121592086931748901383574825, 2.46703708838501200953500525247, 3.39786250515803182705520575721, 5.14564733171739913917542697349, 5.89986463181133977521305231434, 6.30579036571721507470204211305, 7.15582339612203780987701925291, 7.62050216992773679361166762160, 8.838945727981487872032771073719

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