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  − 2.36·2-s − 3-s + 3.61·4-s − 2.39·5-s + 2.36·6-s − 0.116·7-s − 3.82·8-s + 9-s + 5.68·10-s + 6.13·11-s − 3.61·12-s + 13-s + 0.275·14-s + 2.39·15-s + 1.83·16-s + 2.09·17-s − 2.36·18-s − 7.41·19-s − 8.67·20-s + 0.116·21-s − 14.5·22-s − 1.40·23-s + 3.82·24-s + 0.759·25-s − 2.36·26-s − 27-s − 0.420·28-s + ⋯
L(s)  = 1  − 1.67·2-s − 0.577·3-s + 1.80·4-s − 1.07·5-s + 0.967·6-s − 0.0439·7-s − 1.35·8-s + 0.333·9-s + 1.79·10-s + 1.84·11-s − 1.04·12-s + 0.277·13-s + 0.0736·14-s + 0.619·15-s + 0.459·16-s + 0.508·17-s − 0.558·18-s − 1.70·19-s − 1.94·20-s + 0.0253·21-s − 3.09·22-s − 0.293·23-s + 0.781·24-s + 0.151·25-s − 0.464·26-s − 0.192·27-s − 0.0794·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.5462781975$
$L(\frac12)$  $\approx$  $0.5462781975$
$L(\frac{3}{2})$   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 \( 1 + 2.36T + 2T^{2} \)
5 \( 1 + 2.39T + 5T^{2} \)
7 \( 1 + 0.116T + 7T^{2} \)
11 \( 1 - 6.13T + 11T^{2} \)
17 \( 1 - 2.09T + 17T^{2} \)
19 \( 1 + 7.41T + 19T^{2} \)
23 \( 1 + 1.40T + 23T^{2} \)
29 \( 1 - 7.01T + 29T^{2} \)
31 \( 1 - 0.944T + 31T^{2} \)
37 \( 1 - 6.01T + 37T^{2} \)
41 \( 1 - 7.72T + 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 - 2.14T + 53T^{2} \)
59 \( 1 - 5.08T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 8.27T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 + 1.99T + 73T^{2} \)
79 \( 1 - 1.94T + 79T^{2} \)
83 \( 1 + 3.21T + 83T^{2} \)
89 \( 1 + 6.25T + 89T^{2} \)
97 \( 1 + 6.94T + 97T^{2} \)
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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

−8.505677890481735231787416298506, −7.84418341764156081950358273698, −7.17665103975292419496229077950, −6.44110254295602782453146061045, −5.99648386527086009937836464783, −4.26351626770972175729188501037, −4.10909617971784025093709134909, −2.63222561805109971302403831871, −1.39923099691693052571183832432, −0.62314679281945546705286803413, 0.62314679281945546705286803413, 1.39923099691693052571183832432, 2.63222561805109971302403831871, 4.10909617971784025093709134909, 4.26351626770972175729188501037, 5.99648386527086009937836464783, 6.44110254295602782453146061045, 7.17665103975292419496229077950, 7.84418341764156081950358273698, 8.505677890481735231787416298506

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