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  − 1.11·2-s + 3.24·3-s − 0.766·4-s + 5-s − 3.60·6-s + 2.31·7-s + 3.07·8-s + 7.51·9-s − 1.11·10-s + 11-s − 2.48·12-s − 0.104·13-s − 2.57·14-s + 3.24·15-s − 1.87·16-s + 3.59·17-s − 8.34·18-s + 3.68·19-s − 0.766·20-s + 7.51·21-s − 1.11·22-s + 7.11·23-s + 9.96·24-s + 25-s + 0.115·26-s + 14.6·27-s − 1.77·28-s + ⋯
L(s)  = 1  − 0.785·2-s + 1.87·3-s − 0.383·4-s + 0.447·5-s − 1.47·6-s + 0.876·7-s + 1.08·8-s + 2.50·9-s − 0.351·10-s + 0.301·11-s − 0.717·12-s − 0.0289·13-s − 0.687·14-s + 0.837·15-s − 0.469·16-s + 0.873·17-s − 1.96·18-s + 0.846·19-s − 0.171·20-s + 1.64·21-s − 0.236·22-s + 1.48·23-s + 2.03·24-s + 0.200·25-s + 0.0227·26-s + 2.81·27-s − 0.335·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.220487283$
$L(\frac12)$  $\approx$  $3.220487283$
$L(\frac{3}{2})$   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 \( 1 + 1.11T + 2T^{2} \)
3 \( 1 - 3.24T + 3T^{2} \)
7 \( 1 - 2.31T + 7T^{2} \)
13 \( 1 + 0.104T + 13T^{2} \)
17 \( 1 - 3.59T + 17T^{2} \)
19 \( 1 - 3.68T + 19T^{2} \)
23 \( 1 - 7.11T + 23T^{2} \)
29 \( 1 - 4.35T + 29T^{2} \)
31 \( 1 - 0.187T + 31T^{2} \)
37 \( 1 + 2.99T + 37T^{2} \)
41 \( 1 + 9.01T + 41T^{2} \)
43 \( 1 + 5.30T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + 4.26T + 59T^{2} \)
61 \( 1 + 0.802T + 61T^{2} \)
67 \( 1 - 6.84T + 67T^{2} \)
71 \( 1 + 8.85T + 71T^{2} \)
79 \( 1 + 4.89T + 79T^{2} \)
83 \( 1 - 8.39T + 83T^{2} \)
89 \( 1 - 2.19T + 89T^{2} \)
97 \( 1 + 0.769T + 97T^{2} \)
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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

−8.324494147460861072604713799596, −8.145675501839549550617828276857, −7.35732016875580111070555713367, −6.68906575182019331506150878814, −5.06719315100310452879690674160, −4.73328942933591730448546781933, −3.54014558261497007310681176019, −2.97087303139986548138646878883, −1.66725177566375676701605661785, −1.30109536093984588273527779170, 1.30109536093984588273527779170, 1.66725177566375676701605661785, 2.97087303139986548138646878883, 3.54014558261497007310681176019, 4.73328942933591730448546781933, 5.06719315100310452879690674160, 6.68906575182019331506150878814, 7.35732016875580111070555713367, 8.145675501839549550617828276857, 8.324494147460861072604713799596

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