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  + 0.763·2-s + 0.284·3-s − 1.41·4-s + 5-s + 0.217·6-s − 2.19·7-s − 2.60·8-s − 2.91·9-s + 0.763·10-s + 11-s − 0.403·12-s − 0.847·13-s − 1.67·14-s + 0.284·15-s + 0.842·16-s − 3.79·17-s − 2.22·18-s − 3.84·19-s − 1.41·20-s − 0.625·21-s + 0.763·22-s + 5.55·23-s − 0.742·24-s + 25-s − 0.646·26-s − 1.68·27-s + 3.11·28-s + ⋯
L(s)  = 1  + 0.539·2-s + 0.164·3-s − 0.708·4-s + 0.447·5-s + 0.0887·6-s − 0.830·7-s − 0.922·8-s − 0.973·9-s + 0.241·10-s + 0.301·11-s − 0.116·12-s − 0.234·13-s − 0.448·14-s + 0.0734·15-s + 0.210·16-s − 0.920·17-s − 0.525·18-s − 0.882·19-s − 0.316·20-s − 0.136·21-s + 0.162·22-s + 1.15·23-s − 0.151·24-s + 0.200·25-s − 0.126·26-s − 0.324·27-s + 0.588·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.400822951$
$L(\frac12)$  $\approx$  $1.400822951$
$L(\frac{3}{2})$   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 \( 1 - 0.763T + 2T^{2} \)
3 \( 1 - 0.284T + 3T^{2} \)
7 \( 1 + 2.19T + 7T^{2} \)
13 \( 1 + 0.847T + 13T^{2} \)
17 \( 1 + 3.79T + 17T^{2} \)
19 \( 1 + 3.84T + 19T^{2} \)
23 \( 1 - 5.55T + 23T^{2} \)
29 \( 1 + 2.25T + 29T^{2} \)
31 \( 1 + 4.60T + 31T^{2} \)
37 \( 1 - 8.71T + 37T^{2} \)
41 \( 1 - 3.95T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 5.30T + 47T^{2} \)
53 \( 1 - 12.2T + 53T^{2} \)
59 \( 1 + 9.18T + 59T^{2} \)
61 \( 1 - 3.85T + 61T^{2} \)
67 \( 1 + 6.03T + 67T^{2} \)
71 \( 1 + 5.26T + 71T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 2.31T + 83T^{2} \)
89 \( 1 - 16.2T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−8.759944265084888835100228787043, −7.73342977394824467657178151504, −6.76825739450861930855699102517, −6.03040918158669551065964873816, −5.58671834137010184868081732211, −4.58964905824766738332208620257, −3.92111197422178828143998745042, −2.98486522945995560746635773974, −2.33317952892054101086360741836, −0.59424519883884972212965280051, 0.59424519883884972212965280051, 2.33317952892054101086360741836, 2.98486522945995560746635773974, 3.92111197422178828143998745042, 4.58964905824766738332208620257, 5.58671834137010184868081732211, 6.03040918158669551065964873816, 6.76825739450861930855699102517, 7.73342977394824467657178151504, 8.759944265084888835100228787043

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