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  + 2.45·2-s − 3-s + 4.03·4-s + 1.82·5-s − 2.45·6-s + 1.71·7-s + 5.00·8-s + 9-s + 4.48·10-s − 2.47·11-s − 4.03·12-s + 13-s + 4.21·14-s − 1.82·15-s + 4.22·16-s + 2.10·17-s + 2.45·18-s + 5.03·19-s + 7.36·20-s − 1.71·21-s − 6.07·22-s − 4.48·23-s − 5.00·24-s − 1.67·25-s + 2.45·26-s − 27-s + 6.92·28-s + ⋯
L(s)  = 1  + 1.73·2-s − 0.577·3-s + 2.01·4-s + 0.816·5-s − 1.00·6-s + 0.648·7-s + 1.77·8-s + 0.333·9-s + 1.41·10-s − 0.745·11-s − 1.16·12-s + 0.277·13-s + 1.12·14-s − 0.471·15-s + 1.05·16-s + 0.509·17-s + 0.579·18-s + 1.15·19-s + 1.64·20-s − 0.374·21-s − 1.29·22-s − 0.934·23-s − 1.02·24-s − 0.334·25-s + 0.481·26-s − 0.192·27-s + 1.30·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.022151742$
$L(\frac12)$  $\approx$  $6.022151742$
$L(\frac{3}{2})$   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 \( 1 - 2.45T + 2T^{2} \)
5 \( 1 - 1.82T + 5T^{2} \)
7 \( 1 - 1.71T + 7T^{2} \)
11 \( 1 + 2.47T + 11T^{2} \)
17 \( 1 - 2.10T + 17T^{2} \)
19 \( 1 - 5.03T + 19T^{2} \)
23 \( 1 + 4.48T + 23T^{2} \)
29 \( 1 - 2.41T + 29T^{2} \)
31 \( 1 - 8.42T + 31T^{2} \)
37 \( 1 - 5.69T + 37T^{2} \)
41 \( 1 - 11.1T + 41T^{2} \)
43 \( 1 + 2.64T + 43T^{2} \)
47 \( 1 - 6.04T + 47T^{2} \)
53 \( 1 + 1.63T + 53T^{2} \)
59 \( 1 - 2.39T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 7.49T + 67T^{2} \)
71 \( 1 + 1.28T + 71T^{2} \)
73 \( 1 - 4.77T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 5.09T + 83T^{2} \)
89 \( 1 + 5.25T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−8.024236646192742558349667317853, −7.57566888826299498962747707996, −6.50826046348486722501315218938, −5.92550060765860559084831688501, −5.48186067720346040188982451471, −4.77011902367895842028687221623, −4.13358396953775247891446035760, −3.03268208759224528571424252043, −2.29376245696245617961912319732, −1.22807775338954572762789162271, 1.22807775338954572762789162271, 2.29376245696245617961912319732, 3.03268208759224528571424252043, 4.13358396953775247891446035760, 4.77011902367895842028687221623, 5.48186067720346040188982451471, 5.92550060765860559084831688501, 6.50826046348486722501315218938, 7.57566888826299498962747707996, 8.024236646192742558349667317853

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