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  − 2.14·2-s + 2.01·3-s + 2.59·4-s + 5-s − 4.31·6-s − 3.47·7-s − 1.27·8-s + 1.04·9-s − 2.14·10-s + 11-s + 5.21·12-s + 6.08·13-s + 7.44·14-s + 2.01·15-s − 2.45·16-s − 0.860·17-s − 2.24·18-s + 5.32·19-s + 2.59·20-s − 6.98·21-s − 2.14·22-s + 5.95·23-s − 2.56·24-s + 25-s − 13.0·26-s − 3.93·27-s − 9.01·28-s + ⋯
L(s)  = 1  − 1.51·2-s + 1.16·3-s + 1.29·4-s + 0.447·5-s − 1.76·6-s − 1.31·7-s − 0.450·8-s + 0.348·9-s − 0.677·10-s + 0.301·11-s + 1.50·12-s + 1.68·13-s + 1.99·14-s + 0.519·15-s − 0.614·16-s − 0.208·17-s − 0.528·18-s + 1.22·19-s + 0.580·20-s − 1.52·21-s − 0.456·22-s + 1.24·23-s − 0.523·24-s + 0.200·25-s − 2.55·26-s − 0.756·27-s − 1.70·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.416709650$
$L(\frac12)$  $\approx$  $1.416709650$
$L(\frac{3}{2})$   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 \( 1 + 2.14T + 2T^{2} \)
3 \( 1 - 2.01T + 3T^{2} \)
7 \( 1 + 3.47T + 7T^{2} \)
13 \( 1 - 6.08T + 13T^{2} \)
17 \( 1 + 0.860T + 17T^{2} \)
19 \( 1 - 5.32T + 19T^{2} \)
23 \( 1 - 5.95T + 23T^{2} \)
29 \( 1 - 5.45T + 29T^{2} \)
31 \( 1 + 4.31T + 31T^{2} \)
37 \( 1 - 6.22T + 37T^{2} \)
41 \( 1 - 7.88T + 41T^{2} \)
43 \( 1 - 7.05T + 43T^{2} \)
47 \( 1 + 9.39T + 47T^{2} \)
53 \( 1 + 13.5T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 + 7.65T + 61T^{2} \)
67 \( 1 + 4.95T + 67T^{2} \)
71 \( 1 - 9.98T + 71T^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + 15.5T + 83T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + 0.440T + 97T^{2} \)
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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

−8.747448488431643660327226079973, −7.893226810538196103921551708157, −7.32840060706668916262203741302, −6.40102908686957862579901324532, −5.97402518039202897532671968734, −4.48376788963474716814936945618, −3.21624773277632217025073916164, −3.01795833477426070235856243472, −1.73793458717656159253314772533, −0.845783719463800036104152932469, 0.845783719463800036104152932469, 1.73793458717656159253314772533, 3.01795833477426070235856243472, 3.21624773277632217025073916164, 4.48376788963474716814936945618, 5.97402518039202897532671968734, 6.40102908686957862579901324532, 7.32840060706668916262203741302, 7.893226810538196103921551708157, 8.747448488431643660327226079973

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