Properties

Degree 2
Conductor $ 37 \cdot 109 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.67·2-s + 1.31·3-s + 0.808·4-s − 4.46·5-s − 2.21·6-s + 0.153·7-s + 1.99·8-s − 1.26·9-s + 7.47·10-s − 2.02·11-s + 1.06·12-s − 0.641·13-s − 0.257·14-s − 5.88·15-s − 4.96·16-s + 4.82·17-s + 2.11·18-s − 5.72·19-s − 3.60·20-s + 0.202·21-s + 3.39·22-s − 7.85·23-s + 2.63·24-s + 14.9·25-s + 1.07·26-s − 5.61·27-s + 0.124·28-s + ⋯
L(s)  = 1  − 1.18·2-s + 0.761·3-s + 0.404·4-s − 1.99·5-s − 0.902·6-s + 0.0580·7-s + 0.706·8-s − 0.420·9-s + 2.36·10-s − 0.611·11-s + 0.307·12-s − 0.177·13-s − 0.0687·14-s − 1.51·15-s − 1.24·16-s + 1.17·17-s + 0.497·18-s − 1.31·19-s − 0.806·20-s + 0.0441·21-s + 0.724·22-s − 1.63·23-s + 0.537·24-s + 2.98·25-s + 0.210·26-s − 1.08·27-s + 0.0234·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4033\)    =    \(37 \cdot 109\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4033} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4033,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2165291066$
$L(\frac12)$  $\approx$  $0.2165291066$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{37,\;109\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{37,\;109\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad37 \( 1 - T \)
109 \( 1 + T \)
good2 \( 1 + 1.67T + 2T^{2} \)
3 \( 1 - 1.31T + 3T^{2} \)
5 \( 1 + 4.46T + 5T^{2} \)
7 \( 1 - 0.153T + 7T^{2} \)
11 \( 1 + 2.02T + 11T^{2} \)
13 \( 1 + 0.641T + 13T^{2} \)
17 \( 1 - 4.82T + 17T^{2} \)
19 \( 1 + 5.72T + 19T^{2} \)
23 \( 1 + 7.85T + 23T^{2} \)
29 \( 1 - 3.40T + 29T^{2} \)
31 \( 1 + 9.46T + 31T^{2} \)
41 \( 1 + 6.28T + 41T^{2} \)
43 \( 1 + 4.53T + 43T^{2} \)
47 \( 1 - 6.13T + 47T^{2} \)
53 \( 1 - 1.29T + 53T^{2} \)
59 \( 1 + 2.52T + 59T^{2} \)
61 \( 1 + 7.78T + 61T^{2} \)
67 \( 1 + 5.17T + 67T^{2} \)
71 \( 1 - 6.99T + 71T^{2} \)
73 \( 1 + 3.90T + 73T^{2} \)
79 \( 1 - 2.27T + 79T^{2} \)
83 \( 1 - 1.48T + 83T^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
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

−8.217816494576111066104147669600, −8.020950952755982911259159658275, −7.53967837377789433308136340102, −6.68477156096073134317044365956, −5.38723454179377721528972970503, −4.39017594065034101902707335300, −3.76439396444883834145024059917, −2.96057666875337755130776123012, −1.81304480296876577602039263626, −0.29668226578293282696845750006, 0.29668226578293282696845750006, 1.81304480296876577602039263626, 2.96057666875337755130776123012, 3.76439396444883834145024059917, 4.39017594065034101902707335300, 5.38723454179377721528972970503, 6.68477156096073134317044365956, 7.53967837377789433308136340102, 8.020950952755982911259159658275, 8.217816494576111066104147669600

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