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  + 2.69·2-s − 3-s + 5.23·4-s − 3.39·5-s − 2.69·6-s + 3.55·7-s + 8.71·8-s + 9-s − 9.12·10-s + 3.97·11-s − 5.23·12-s + 13-s + 9.57·14-s + 3.39·15-s + 12.9·16-s − 5.69·17-s + 2.69·18-s − 3.19·19-s − 17.7·20-s − 3.55·21-s + 10.6·22-s + 4.70·23-s − 8.71·24-s + 6.49·25-s + 2.69·26-s − 27-s + 18.6·28-s + ⋯
L(s)  = 1  + 1.90·2-s − 0.577·3-s + 2.61·4-s − 1.51·5-s − 1.09·6-s + 1.34·7-s + 3.08·8-s + 0.333·9-s − 2.88·10-s + 1.19·11-s − 1.51·12-s + 0.277·13-s + 2.55·14-s + 0.875·15-s + 3.24·16-s − 1.38·17-s + 0.634·18-s − 0.732·19-s − 3.97·20-s − 0.776·21-s + 2.27·22-s + 0.981·23-s − 1.77·24-s + 1.29·25-s + 0.527·26-s − 0.192·27-s + 3.52·28-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  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.513261643$
$L(\frac12)$  $\approx$  $5.513261643$
$L(\frac{3}{2})$   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 \( 1 - 2.69T + 2T^{2} \)
5 \( 1 + 3.39T + 5T^{2} \)
7 \( 1 - 3.55T + 7T^{2} \)
11 \( 1 - 3.97T + 11T^{2} \)
17 \( 1 + 5.69T + 17T^{2} \)
19 \( 1 + 3.19T + 19T^{2} \)
23 \( 1 - 4.70T + 23T^{2} \)
29 \( 1 - 6.05T + 29T^{2} \)
31 \( 1 - 6.73T + 31T^{2} \)
37 \( 1 + 7.13T + 37T^{2} \)
41 \( 1 - 1.26T + 41T^{2} \)
43 \( 1 + 3.71T + 43T^{2} \)
47 \( 1 + 2.79T + 47T^{2} \)
53 \( 1 - 14.3T + 53T^{2} \)
59 \( 1 - 10.9T + 59T^{2} \)
61 \( 1 + 9.35T + 61T^{2} \)
67 \( 1 + 8.20T + 67T^{2} \)
71 \( 1 + 5.77T + 71T^{2} \)
73 \( 1 + 4.07T + 73T^{2} \)
79 \( 1 - 9.95T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 - 14.6T + 89T^{2} \)
97 \( 1 - 6.18T + 97T^{2} \)
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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

−8.253261089493115356373578235155, −7.30737618362928527584654129270, −6.76905066032857596468060286023, −6.22628846608848718154210062480, −5.04206689921684264405724170933, −4.59527374105784617472495169624, −4.16282851578012103481679865714, −3.42166353250805979118557137991, −2.23210601427318104162047514129, −1.13183107888023680539055164761, 1.13183107888023680539055164761, 2.23210601427318104162047514129, 3.42166353250805979118557137991, 4.16282851578012103481679865714, 4.59527374105784617472495169624, 5.04206689921684264405724170933, 6.22628846608848718154210062480, 6.76905066032857596468060286023, 7.30737618362928527584654129270, 8.253261089493115356373578235155

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