Properties

Degree 2
Conductor $ 5 \cdot 11 \cdot 73 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.72·2-s − 1.60·3-s + 5.44·4-s + 5-s − 4.38·6-s + 0.0202·7-s + 9.39·8-s − 0.414·9-s + 2.72·10-s + 11-s − 8.75·12-s − 5.65·13-s + 0.0552·14-s − 1.60·15-s + 14.7·16-s + 2.53·17-s − 1.13·18-s + 3.30·19-s + 5.44·20-s − 0.0325·21-s + 2.72·22-s + 3.06·23-s − 15.1·24-s + 25-s − 15.4·26-s + 5.49·27-s + 0.110·28-s + ⋯
L(s)  = 1  + 1.92·2-s − 0.928·3-s + 2.72·4-s + 0.447·5-s − 1.79·6-s + 0.00765·7-s + 3.32·8-s − 0.138·9-s + 0.862·10-s + 0.301·11-s − 2.52·12-s − 1.56·13-s + 0.0147·14-s − 0.415·15-s + 3.68·16-s + 0.614·17-s − 0.266·18-s + 0.757·19-s + 1.21·20-s − 0.00710·21-s + 0.581·22-s + 0.638·23-s − 3.08·24-s + 0.200·25-s − 3.02·26-s + 1.05·27-s + 0.0208·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4015\)    =    \(5 \cdot 11 \cdot 73\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4015} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.604484955$
$L(\frac12)$  $\approx$  $5.604484955$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11,\;73\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;11,\;73\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 - T \)
11 \( 1 - T \)
73 \( 1 - T \)
good2 \( 1 - 2.72T + 2T^{2} \)
3 \( 1 + 1.60T + 3T^{2} \)
7 \( 1 - 0.0202T + 7T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 - 2.53T + 17T^{2} \)
19 \( 1 - 3.30T + 19T^{2} \)
23 \( 1 - 3.06T + 23T^{2} \)
29 \( 1 - 0.877T + 29T^{2} \)
31 \( 1 - 7.95T + 31T^{2} \)
37 \( 1 + 10.5T + 37T^{2} \)
41 \( 1 - 9.06T + 41T^{2} \)
43 \( 1 - 9.08T + 43T^{2} \)
47 \( 1 - 7.34T + 47T^{2} \)
53 \( 1 + 0.627T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 9.37T + 61T^{2} \)
67 \( 1 - 2.36T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
79 \( 1 + 1.64T + 79T^{2} \)
83 \( 1 + 14.6T + 83T^{2} \)
89 \( 1 + 1.29T + 89T^{2} \)
97 \( 1 - 4.49T + 97T^{2} \)
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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

−8.086307203498831486142067758301, −7.10599637944665738482649819435, −6.81882529250860656191776906635, −5.81263217941647318180975559208, −5.45795250110575218547716040219, −4.86625244803067626418108874461, −4.12842387086101594470857310008, −2.97652040091792542573039452704, −2.47061369420225371899463897067, −1.12242515787014427944251115599, 1.12242515787014427944251115599, 2.47061369420225371899463897067, 2.97652040091792542573039452704, 4.12842387086101594470857310008, 4.86625244803067626418108874461, 5.45795250110575218547716040219, 5.81263217941647318180975559208, 6.81882529250860656191776906635, 7.10599637944665738482649819435, 8.086307203498831486142067758301

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