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  − 2.41·2-s − 3.18·3-s + 3.81·4-s + 5-s + 7.67·6-s − 3.73·7-s − 4.37·8-s + 7.11·9-s − 2.41·10-s + 11-s − 12.1·12-s + 1.00·13-s + 9.01·14-s − 3.18·15-s + 2.92·16-s + 3.53·17-s − 17.1·18-s + 5.00·19-s + 3.81·20-s + 11.8·21-s − 2.41·22-s − 0.391·23-s + 13.9·24-s + 25-s − 2.42·26-s − 13.1·27-s − 14.2·28-s + ⋯
L(s)  = 1  − 1.70·2-s − 1.83·3-s + 1.90·4-s + 0.447·5-s + 3.13·6-s − 1.41·7-s − 1.54·8-s + 2.37·9-s − 0.762·10-s + 0.301·11-s − 3.50·12-s + 0.279·13-s + 2.40·14-s − 0.821·15-s + 0.730·16-s + 0.858·17-s − 4.04·18-s + 1.14·19-s + 0.853·20-s + 2.59·21-s − 0.514·22-s − 0.0816·23-s + 2.84·24-s + 0.200·25-s − 0.476·26-s − 2.52·27-s − 2.69·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4409068701$
$L(\frac12)$  $\approx$  $0.4409068701$
$L(\frac{3}{2})$   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 \( 1 + 2.41T + 2T^{2} \)
3 \( 1 + 3.18T + 3T^{2} \)
7 \( 1 + 3.73T + 7T^{2} \)
13 \( 1 - 1.00T + 13T^{2} \)
17 \( 1 - 3.53T + 17T^{2} \)
19 \( 1 - 5.00T + 19T^{2} \)
23 \( 1 + 0.391T + 23T^{2} \)
29 \( 1 - 5.23T + 29T^{2} \)
31 \( 1 - 0.906T + 31T^{2} \)
37 \( 1 - 9.23T + 37T^{2} \)
41 \( 1 + 3.85T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 5.91T + 47T^{2} \)
53 \( 1 - 9.74T + 53T^{2} \)
59 \( 1 - 8.90T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 5.52T + 67T^{2} \)
71 \( 1 + 8.76T + 71T^{2} \)
79 \( 1 + 5.30T + 79T^{2} \)
83 \( 1 - 5.31T + 83T^{2} \)
89 \( 1 - 7.61T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−8.597215922892708363721724532340, −7.51709125515896154373561810057, −7.00790317167696014115085384368, −6.27791199848613951268805191188, −5.95399076775963889947252045894, −5.04024774133025985607384368672, −3.78655911613187137098367425008, −2.58330331046380179498626359257, −1.16190240931778392829843440145, −0.66690340992398613091760487779, 0.66690340992398613091760487779, 1.16190240931778392829843440145, 2.58330331046380179498626359257, 3.78655911613187137098367425008, 5.04024774133025985607384368672, 5.95399076775963889947252045894, 6.27791199848613951268805191188, 7.00790317167696014115085384368, 7.51709125515896154373561810057, 8.597215922892708363721724532340

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