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  + 2.66·2-s + 1.84·3-s + 5.12·4-s + 5-s + 4.92·6-s + 0.877·7-s + 8.33·8-s + 0.406·9-s + 2.66·10-s + 11-s + 9.45·12-s − 1.10·13-s + 2.34·14-s + 1.84·15-s + 11.9·16-s − 5.26·17-s + 1.08·18-s + 1.15·19-s + 5.12·20-s + 1.61·21-s + 2.66·22-s + 1.08·23-s + 15.3·24-s + 25-s − 2.93·26-s − 4.78·27-s + 4.49·28-s + ⋯
L(s)  = 1  + 1.88·2-s + 1.06·3-s + 2.56·4-s + 0.447·5-s + 2.01·6-s + 0.331·7-s + 2.94·8-s + 0.135·9-s + 0.843·10-s + 0.301·11-s + 2.72·12-s − 0.305·13-s + 0.625·14-s + 0.476·15-s + 2.99·16-s − 1.27·17-s + 0.255·18-s + 0.264·19-s + 1.14·20-s + 0.353·21-s + 0.569·22-s + 0.226·23-s + 3.13·24-s + 0.200·25-s − 0.576·26-s − 0.921·27-s + 0.849·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $10.25739912$
$L(\frac12)$  $\approx$  $10.25739912$
$L(\frac{3}{2})$   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 \( 1 - 2.66T + 2T^{2} \)
3 \( 1 - 1.84T + 3T^{2} \)
7 \( 1 - 0.877T + 7T^{2} \)
13 \( 1 + 1.10T + 13T^{2} \)
17 \( 1 + 5.26T + 17T^{2} \)
19 \( 1 - 1.15T + 19T^{2} \)
23 \( 1 - 1.08T + 23T^{2} \)
29 \( 1 - 3.24T + 29T^{2} \)
31 \( 1 - 3.80T + 31T^{2} \)
37 \( 1 - 1.48T + 37T^{2} \)
41 \( 1 + 11.6T + 41T^{2} \)
43 \( 1 + 3.34T + 43T^{2} \)
47 \( 1 - 4.90T + 47T^{2} \)
53 \( 1 + 6.99T + 53T^{2} \)
59 \( 1 + 5.29T + 59T^{2} \)
61 \( 1 - 8.99T + 61T^{2} \)
67 \( 1 + 6.29T + 67T^{2} \)
71 \( 1 - 5.97T + 71T^{2} \)
79 \( 1 + 1.71T + 79T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 - 6.20T + 89T^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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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

−8.305135740894903568475833689383, −7.53519514619010597536661345031, −6.67052568370162936698413979140, −6.24005359199289768995506842426, −5.16420798380322468575594015439, −4.70038368046815319989682096456, −3.80680091914166640931117579323, −3.06408133976634759477987106463, −2.38653626136514610368527013841, −1.66373463972223431084694191705, 1.66373463972223431084694191705, 2.38653626136514610368527013841, 3.06408133976634759477987106463, 3.80680091914166640931117579323, 4.70038368046815319989682096456, 5.16420798380322468575594015439, 6.24005359199289768995506842426, 6.67052568370162936698413979140, 7.53519514619010597536661345031, 8.305135740894903568475833689383

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