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  − 1.79·2-s − 3-s + 1.23·4-s + 1.64·5-s + 1.79·6-s + 1.60·7-s + 1.37·8-s + 9-s − 2.96·10-s − 1.34·11-s − 1.23·12-s + 13-s − 2.89·14-s − 1.64·15-s − 4.94·16-s − 5.68·17-s − 1.79·18-s − 1.69·19-s + 2.03·20-s − 1.60·21-s + 2.42·22-s − 1.20·23-s − 1.37·24-s − 2.28·25-s − 1.79·26-s − 27-s + 1.99·28-s + ⋯
L(s)  = 1  − 1.27·2-s − 0.577·3-s + 0.618·4-s + 0.736·5-s + 0.734·6-s + 0.608·7-s + 0.485·8-s + 0.333·9-s − 0.937·10-s − 0.406·11-s − 0.357·12-s + 0.277·13-s − 0.774·14-s − 0.425·15-s − 1.23·16-s − 1.37·17-s − 0.424·18-s − 0.388·19-s + 0.455·20-s − 0.351·21-s + 0.517·22-s − 0.250·23-s − 0.280·24-s − 0.457·25-s − 0.352·26-s − 0.192·27-s + 0.376·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.7405981032$
$L(\frac12)$  $\approx$  $0.7405981032$
$L(\frac{3}{2})$   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 \( 1 + 1.79T + 2T^{2} \)
5 \( 1 - 1.64T + 5T^{2} \)
7 \( 1 - 1.60T + 7T^{2} \)
11 \( 1 + 1.34T + 11T^{2} \)
17 \( 1 + 5.68T + 17T^{2} \)
19 \( 1 + 1.69T + 19T^{2} \)
23 \( 1 + 1.20T + 23T^{2} \)
29 \( 1 - 4.18T + 29T^{2} \)
31 \( 1 + 9.37T + 31T^{2} \)
37 \( 1 + 0.916T + 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 4.44T + 47T^{2} \)
53 \( 1 + 6.29T + 53T^{2} \)
59 \( 1 + 0.514T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 - 3.26T + 67T^{2} \)
71 \( 1 - 8.99T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 - 6.11T + 79T^{2} \)
83 \( 1 - 0.847T + 83T^{2} \)
89 \( 1 + 1.04T + 89T^{2} \)
97 \( 1 + 1.73T + 97T^{2} \)
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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

−8.601452111354309393753227394045, −7.79054673241931382396683601271, −7.17761115383874095133320615465, −6.32482775624907100869511275812, −5.63317256990373997432950853411, −4.73390874610790048494524601453, −4.04486451234456195678271611784, −2.35880467170253295499071349348, −1.79002211825901027412872549772, −0.61932687048761507787696553104, 0.61932687048761507787696553104, 1.79002211825901027412872549772, 2.35880467170253295499071349348, 4.04486451234456195678271611784, 4.73390874610790048494524601453, 5.63317256990373997432950853411, 6.32482775624907100869511275812, 7.17761115383874095133320615465, 7.79054673241931382396683601271, 8.601452111354309393753227394045

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