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  + 0.503·2-s + 0.0692·3-s − 1.74·4-s + 5-s + 0.0348·6-s − 2.92·7-s − 1.88·8-s − 2.99·9-s + 0.503·10-s + 11-s − 0.120·12-s + 5.73·13-s − 1.47·14-s + 0.0692·15-s + 2.54·16-s + 0.148·17-s − 1.50·18-s − 4.90·19-s − 1.74·20-s − 0.202·21-s + 0.503·22-s − 6.36·23-s − 0.130·24-s + 25-s + 2.88·26-s − 0.414·27-s + 5.10·28-s + ⋯
L(s)  = 1  + 0.356·2-s + 0.0399·3-s − 0.873·4-s + 0.447·5-s + 0.0142·6-s − 1.10·7-s − 0.666·8-s − 0.998·9-s + 0.159·10-s + 0.301·11-s − 0.0348·12-s + 1.59·13-s − 0.393·14-s + 0.0178·15-s + 0.635·16-s + 0.0359·17-s − 0.355·18-s − 1.12·19-s − 0.390·20-s − 0.0441·21-s + 0.107·22-s − 1.32·23-s − 0.0266·24-s + 0.200·25-s + 0.566·26-s − 0.0798·27-s + 0.964·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.242811945$
$L(\frac12)$  $\approx$  $1.242811945$
$L(\frac{3}{2})$   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 \( 1 - 0.503T + 2T^{2} \)
3 \( 1 - 0.0692T + 3T^{2} \)
7 \( 1 + 2.92T + 7T^{2} \)
13 \( 1 - 5.73T + 13T^{2} \)
17 \( 1 - 0.148T + 17T^{2} \)
19 \( 1 + 4.90T + 19T^{2} \)
23 \( 1 + 6.36T + 23T^{2} \)
29 \( 1 + 1.48T + 29T^{2} \)
31 \( 1 - 6.03T + 31T^{2} \)
37 \( 1 + 6.02T + 37T^{2} \)
41 \( 1 - 1.06T + 41T^{2} \)
43 \( 1 - 0.407T + 43T^{2} \)
47 \( 1 + 3.78T + 47T^{2} \)
53 \( 1 - 0.702T + 53T^{2} \)
59 \( 1 - 4.46T + 59T^{2} \)
61 \( 1 + 2.17T + 61T^{2} \)
67 \( 1 - 4.71T + 67T^{2} \)
71 \( 1 - 6.20T + 71T^{2} \)
79 \( 1 + 7.70T + 79T^{2} \)
83 \( 1 - 15.6T + 83T^{2} \)
89 \( 1 - 3.63T + 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−8.639510245333389787579964615830, −7.979934478793526060529250077121, −6.52095055089646332047440969736, −6.19139203042901174934705615484, −5.65757547497701490837077402876, −4.61969101499899269250864651949, −3.68545871946216631999052125434, −3.29219211482036219881127153205, −2.09197857074856161020224790320, −0.58928345151238958919741372708, 0.58928345151238958919741372708, 2.09197857074856161020224790320, 3.29219211482036219881127153205, 3.68545871946216631999052125434, 4.61969101499899269250864651949, 5.65757547497701490837077402876, 6.19139203042901174934705615484, 6.52095055089646332047440969736, 7.979934478793526060529250077121, 8.639510245333389787579964615830

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