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  − 0.446·2-s + 1.08·3-s − 1.80·4-s + 5-s − 0.484·6-s + 4.20·7-s + 1.69·8-s − 1.82·9-s − 0.446·10-s + 11-s − 1.95·12-s − 2.82·13-s − 1.87·14-s + 1.08·15-s + 2.84·16-s + 4.54·17-s + 0.814·18-s − 4.76·19-s − 1.80·20-s + 4.56·21-s − 0.446·22-s + 2.76·23-s + 1.84·24-s + 25-s + 1.26·26-s − 5.23·27-s − 7.56·28-s + ⋯
L(s)  = 1  − 0.315·2-s + 0.626·3-s − 0.900·4-s + 0.447·5-s − 0.197·6-s + 1.58·7-s + 0.600·8-s − 0.607·9-s − 0.141·10-s + 0.301·11-s − 0.563·12-s − 0.783·13-s − 0.501·14-s + 0.280·15-s + 0.710·16-s + 1.10·17-s + 0.191·18-s − 1.09·19-s − 0.402·20-s + 0.995·21-s − 0.0952·22-s + 0.576·23-s + 0.375·24-s + 0.200·25-s + 0.247·26-s − 1.00·27-s − 1.43·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.082829485$
$L(\frac12)$  $\approx$  $2.082829485$
$L(\frac{3}{2})$   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 \( 1 + 0.446T + 2T^{2} \)
3 \( 1 - 1.08T + 3T^{2} \)
7 \( 1 - 4.20T + 7T^{2} \)
13 \( 1 + 2.82T + 13T^{2} \)
17 \( 1 - 4.54T + 17T^{2} \)
19 \( 1 + 4.76T + 19T^{2} \)
23 \( 1 - 2.76T + 23T^{2} \)
29 \( 1 + 2.06T + 29T^{2} \)
31 \( 1 - 7.10T + 31T^{2} \)
37 \( 1 + 5.06T + 37T^{2} \)
41 \( 1 - 9.63T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + 0.673T + 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 + 14.0T + 59T^{2} \)
61 \( 1 - 1.12T + 61T^{2} \)
67 \( 1 - 7.73T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
79 \( 1 + 3.88T + 79T^{2} \)
83 \( 1 - 3.76T + 83T^{2} \)
89 \( 1 - 3.77T + 89T^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−8.730930891227815522390672791281, −7.72599713146430607093204548444, −7.51924048470391281993935758126, −6.05826854588228804623097231623, −5.35656560015349279271131402717, −4.66758786869669262199507988172, −3.98810940925418748162478202981, −2.80398767761468826654804414143, −1.92579255203695281742361661946, −0.885860926227460055264406354070, 0.885860926227460055264406354070, 1.92579255203695281742361661946, 2.80398767761468826654804414143, 3.98810940925418748162478202981, 4.66758786869669262199507988172, 5.35656560015349279271131402717, 6.05826854588228804623097231623, 7.51924048470391281993935758126, 7.72599713146430607093204548444, 8.730930891227815522390672791281

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