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  + 0.984·2-s + 3.01·3-s − 1.03·4-s + 5-s + 2.96·6-s + 3.40·7-s − 2.98·8-s + 6.09·9-s + 0.984·10-s + 11-s − 3.10·12-s + 3.41·13-s + 3.35·14-s + 3.01·15-s − 0.877·16-s − 1.53·17-s + 5.99·18-s − 4.91·19-s − 1.03·20-s + 10.2·21-s + 0.984·22-s + 1.63·23-s − 8.99·24-s + 25-s + 3.36·26-s + 9.31·27-s − 3.50·28-s + ⋯
L(s)  = 1  + 0.696·2-s + 1.74·3-s − 0.515·4-s + 0.447·5-s + 1.21·6-s + 1.28·7-s − 1.05·8-s + 2.03·9-s + 0.311·10-s + 0.301·11-s − 0.896·12-s + 0.947·13-s + 0.896·14-s + 0.778·15-s − 0.219·16-s − 0.371·17-s + 1.41·18-s − 1.12·19-s − 0.230·20-s + 2.24·21-s + 0.209·22-s + 0.341·23-s − 1.83·24-s + 0.200·25-s + 0.659·26-s + 1.79·27-s − 0.663·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.726645185$
$L(\frac12)$  $\approx$  $5.726645185$
$L(\frac{3}{2})$   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 \( 1 - 0.984T + 2T^{2} \)
3 \( 1 - 3.01T + 3T^{2} \)
7 \( 1 - 3.40T + 7T^{2} \)
13 \( 1 - 3.41T + 13T^{2} \)
17 \( 1 + 1.53T + 17T^{2} \)
19 \( 1 + 4.91T + 19T^{2} \)
23 \( 1 - 1.63T + 23T^{2} \)
29 \( 1 - 0.688T + 29T^{2} \)
31 \( 1 - 0.0844T + 31T^{2} \)
37 \( 1 + 5.26T + 37T^{2} \)
41 \( 1 - 1.21T + 41T^{2} \)
43 \( 1 - 4.68T + 43T^{2} \)
47 \( 1 - 10.2T + 47T^{2} \)
53 \( 1 + 0.878T + 53T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 - 3.14T + 61T^{2} \)
67 \( 1 + 2.82T + 67T^{2} \)
71 \( 1 - 5.36T + 71T^{2} \)
79 \( 1 - 1.01T + 79T^{2} \)
83 \( 1 + 4.02T + 83T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + 5.29T + 97T^{2} \)
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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

−8.509514573312491416112088413056, −8.038188804312560617838348419944, −7.07290231324557286146396610079, −6.16659762718305680499077998125, −5.24901355040005012536519390577, −4.32814247546025511910544021883, −4.00611671541923771615333696804, −3.02232919073635727724599437559, −2.18639895320115787963870745776, −1.31792833822744280889477911565, 1.31792833822744280889477911565, 2.18639895320115787963870745776, 3.02232919073635727724599437559, 4.00611671541923771615333696804, 4.32814247546025511910544021883, 5.24901355040005012536519390577, 6.16659762718305680499077998125, 7.07290231324557286146396610079, 8.038188804312560617838348419944, 8.509514573312491416112088413056

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