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  + 0.506·2-s − 3-s − 1.74·4-s + 4.00·5-s − 0.506·6-s + 4.25·7-s − 1.89·8-s + 9-s + 2.02·10-s − 0.00859·11-s + 1.74·12-s + 13-s + 2.15·14-s − 4.00·15-s + 2.52·16-s − 2.55·17-s + 0.506·18-s + 5.31·19-s − 6.98·20-s − 4.25·21-s − 0.00435·22-s + 7.33·23-s + 1.89·24-s + 11.0·25-s + 0.506·26-s − 27-s − 7.41·28-s + ⋯
L(s)  = 1  + 0.358·2-s − 0.577·3-s − 0.871·4-s + 1.79·5-s − 0.206·6-s + 1.60·7-s − 0.670·8-s + 0.333·9-s + 0.641·10-s − 0.00259·11-s + 0.503·12-s + 0.277·13-s + 0.575·14-s − 1.03·15-s + 0.631·16-s − 0.619·17-s + 0.119·18-s + 1.21·19-s − 1.56·20-s − 0.927·21-s − 0.000927·22-s + 1.52·23-s + 0.387·24-s + 2.21·25-s + 0.0993·26-s − 0.192·27-s − 1.40·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.828025078$
$L(\frac12)$  $\approx$  $2.828025078$
$L(\frac{3}{2})$   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 \( 1 - 0.506T + 2T^{2} \)
5 \( 1 - 4.00T + 5T^{2} \)
7 \( 1 - 4.25T + 7T^{2} \)
11 \( 1 + 0.00859T + 11T^{2} \)
17 \( 1 + 2.55T + 17T^{2} \)
19 \( 1 - 5.31T + 19T^{2} \)
23 \( 1 - 7.33T + 23T^{2} \)
29 \( 1 + 3.63T + 29T^{2} \)
31 \( 1 + 4.47T + 31T^{2} \)
37 \( 1 + 4.40T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 + 8.16T + 43T^{2} \)
47 \( 1 - 10.6T + 47T^{2} \)
53 \( 1 + 8.47T + 53T^{2} \)
59 \( 1 + 9.30T + 59T^{2} \)
61 \( 1 - 9.90T + 61T^{2} \)
67 \( 1 + 4.26T + 67T^{2} \)
71 \( 1 + 5.41T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 + 10.8T + 79T^{2} \)
83 \( 1 - 4.73T + 83T^{2} \)
89 \( 1 - 14.1T + 89T^{2} \)
97 \( 1 + 16.4T + 97T^{2} \)
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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

−8.666518751454620755830704807707, −7.68558302654070061165641393076, −6.82964501223126358654937411708, −5.88139658841831420730834291770, −5.26391089358505282328059127592, −5.07603811252372072124590176374, −4.15102058374365962613916853401, −2.89754499579027140753416316087, −1.76251057236841040416187344618, −1.06090014014777302345335766748, 1.06090014014777302345335766748, 1.76251057236841040416187344618, 2.89754499579027140753416316087, 4.15102058374365962613916853401, 5.07603811252372072124590176374, 5.26391089358505282328059127592, 5.88139658841831420730834291770, 6.82964501223126358654937411708, 7.68558302654070061165641393076, 8.666518751454620755830704807707

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