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  + 0.829·2-s + 2.15·3-s − 1.31·4-s + 5-s + 1.78·6-s − 1.04·7-s − 2.74·8-s + 1.63·9-s + 0.829·10-s + 11-s − 2.82·12-s − 4.99·13-s − 0.865·14-s + 2.15·15-s + 0.341·16-s + 2.47·17-s + 1.35·18-s + 2.20·19-s − 1.31·20-s − 2.24·21-s + 0.829·22-s + 5.73·23-s − 5.91·24-s + 25-s − 4.14·26-s − 2.94·27-s + 1.36·28-s + ⋯
L(s)  = 1  + 0.586·2-s + 1.24·3-s − 0.655·4-s + 0.447·5-s + 0.729·6-s − 0.394·7-s − 0.971·8-s + 0.544·9-s + 0.262·10-s + 0.301·11-s − 0.814·12-s − 1.38·13-s − 0.231·14-s + 0.555·15-s + 0.0854·16-s + 0.601·17-s + 0.319·18-s + 0.506·19-s − 0.293·20-s − 0.489·21-s + 0.176·22-s + 1.19·23-s − 1.20·24-s + 0.200·25-s − 0.813·26-s − 0.565·27-s + 0.258·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.326714339$
$L(\frac12)$  $\approx$  $3.326714339$
$L(\frac{3}{2})$   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 \( 1 - 0.829T + 2T^{2} \)
3 \( 1 - 2.15T + 3T^{2} \)
7 \( 1 + 1.04T + 7T^{2} \)
13 \( 1 + 4.99T + 13T^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
19 \( 1 - 2.20T + 19T^{2} \)
23 \( 1 - 5.73T + 23T^{2} \)
29 \( 1 - 7.55T + 29T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 + 2.81T + 37T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 - 6.23T + 43T^{2} \)
47 \( 1 + 7.66T + 47T^{2} \)
53 \( 1 + 4.99T + 53T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + 1.05T + 61T^{2} \)
67 \( 1 - 7.00T + 67T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
79 \( 1 - 15.2T + 79T^{2} \)
83 \( 1 - 3.03T + 83T^{2} \)
89 \( 1 + 16.0T + 89T^{2} \)
97 \( 1 - 2.92T + 97T^{2} \)
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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

−8.433581757954416421770310839337, −7.936260476473249371071600979851, −6.94793359188713571904188417555, −6.23913667574727684641283547274, −5.17166655711424069392820367564, −4.73213487996104821164134225991, −3.69085522242683386270857908058, −2.95130617047330262038183769421, −2.48487168498962943018554784147, −0.916351500627204995195981911081, 0.916351500627204995195981911081, 2.48487168498962943018554784147, 2.95130617047330262038183769421, 3.69085522242683386270857908058, 4.73213487996104821164134225991, 5.17166655711424069392820367564, 6.23913667574727684641283547274, 6.94793359188713571904188417555, 7.936260476473249371071600979851, 8.433581757954416421770310839337

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