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  + 0.877·2-s − 3.19·3-s − 1.22·4-s + 5-s − 2.80·6-s + 3.15·7-s − 2.83·8-s + 7.20·9-s + 0.877·10-s + 11-s + 3.92·12-s − 0.594·13-s + 2.77·14-s − 3.19·15-s − 0.0275·16-s + 7.12·17-s + 6.32·18-s + 4.38·19-s − 1.22·20-s − 10.0·21-s + 0.877·22-s − 3.31·23-s + 9.05·24-s + 25-s − 0.521·26-s − 13.4·27-s − 3.88·28-s + ⋯
L(s)  = 1  + 0.620·2-s − 1.84·3-s − 0.614·4-s + 0.447·5-s − 1.14·6-s + 1.19·7-s − 1.00·8-s + 2.40·9-s + 0.277·10-s + 0.301·11-s + 1.13·12-s − 0.164·13-s + 0.741·14-s − 0.824·15-s − 0.00688·16-s + 1.72·17-s + 1.49·18-s + 1.00·19-s − 0.275·20-s − 2.20·21-s + 0.187·22-s − 0.691·23-s + 1.84·24-s + 0.200·25-s − 0.102·26-s − 2.58·27-s − 0.734·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.540291021$
$L(\frac12)$  $\approx$  $1.540291021$
$L(\frac{3}{2})$   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 \( 1 - 0.877T + 2T^{2} \)
3 \( 1 + 3.19T + 3T^{2} \)
7 \( 1 - 3.15T + 7T^{2} \)
13 \( 1 + 0.594T + 13T^{2} \)
17 \( 1 - 7.12T + 17T^{2} \)
19 \( 1 - 4.38T + 19T^{2} \)
23 \( 1 + 3.31T + 23T^{2} \)
29 \( 1 - 9.46T + 29T^{2} \)
31 \( 1 + 6.81T + 31T^{2} \)
37 \( 1 + 0.491T + 37T^{2} \)
41 \( 1 - 12.5T + 41T^{2} \)
43 \( 1 + 1.86T + 43T^{2} \)
47 \( 1 + 8.15T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 + 3.46T + 59T^{2} \)
61 \( 1 - 0.193T + 61T^{2} \)
67 \( 1 - 12.0T + 67T^{2} \)
71 \( 1 + 3.50T + 71T^{2} \)
79 \( 1 + 0.753T + 79T^{2} \)
83 \( 1 + 8.54T + 83T^{2} \)
89 \( 1 + 2.93T + 89T^{2} \)
97 \( 1 - 6.06T + 97T^{2} \)
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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

−8.256233616477786610389173380706, −7.62406882528575646407979456258, −6.67093929760570942717568775939, −5.80934546939345503501539742961, −5.50022827493204674174866147334, −4.84180516970295568884713736009, −4.31776223465101690354668111911, −3.23607125239294011725907960357, −1.57183834334289426594127931973, −0.790367246908459902112496794168, 0.790367246908459902112496794168, 1.57183834334289426594127931973, 3.23607125239294011725907960357, 4.31776223465101690354668111911, 4.84180516970295568884713736009, 5.50022827493204674174866147334, 5.80934546939345503501539742961, 6.67093929760570942717568775939, 7.62406882528575646407979456258, 8.256233616477786610389173380706

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