Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 503 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 0.0581·5-s − 2.78·7-s + 9-s + 3.35·11-s − 1.13·13-s − 0.0581·15-s + 1.84·17-s − 0.820·19-s + 2.78·21-s − 4.45·23-s − 4.99·25-s − 27-s + 2.16·29-s − 4.12·31-s − 3.35·33-s − 0.161·35-s + 2.83·37-s + 1.13·39-s + 11.8·41-s − 2.14·43-s + 0.0581·45-s + 2.54·47-s + 0.745·49-s − 1.84·51-s + 9.14·53-s + 0.195·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.0259·5-s − 1.05·7-s + 0.333·9-s + 1.01·11-s − 0.316·13-s − 0.0150·15-s + 0.447·17-s − 0.188·19-s + 0.607·21-s − 0.929·23-s − 0.999·25-s − 0.192·27-s + 0.401·29-s − 0.740·31-s − 0.584·33-s − 0.0273·35-s + 0.465·37-s + 0.182·39-s + 1.85·41-s − 0.326·43-s + 0.00866·45-s + 0.371·47-s + 0.106·49-s − 0.258·51-s + 1.25·53-s + 0.0263·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6036\)    =    \(2^{2} \cdot 3 \cdot 503\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{6036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.153704994$
$L(\frac12)$  $\approx$  $1.153704994$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;503\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;503\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
503 \( 1 + T \)
good5 \( 1 - 0.0581T + 5T^{2} \)
7 \( 1 + 2.78T + 7T^{2} \)
11 \( 1 - 3.35T + 11T^{2} \)
13 \( 1 + 1.13T + 13T^{2} \)
17 \( 1 - 1.84T + 17T^{2} \)
19 \( 1 + 0.820T + 19T^{2} \)
23 \( 1 + 4.45T + 23T^{2} \)
29 \( 1 - 2.16T + 29T^{2} \)
31 \( 1 + 4.12T + 31T^{2} \)
37 \( 1 - 2.83T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 + 2.14T + 43T^{2} \)
47 \( 1 - 2.54T + 47T^{2} \)
53 \( 1 - 9.14T + 53T^{2} \)
59 \( 1 + 5.43T + 59T^{2} \)
61 \( 1 + 3.80T + 61T^{2} \)
67 \( 1 + 8.54T + 67T^{2} \)
71 \( 1 - 7.66T + 71T^{2} \)
73 \( 1 - 6.97T + 73T^{2} \)
79 \( 1 - 3.92T + 79T^{2} \)
83 \( 1 + 7.00T + 83T^{2} \)
89 \( 1 + 0.190T + 89T^{2} \)
97 \( 1 + 5.39T + 97T^{2} \)
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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

−7.919407638442506951583073500571, −7.30423973325718076760557733814, −6.48099804983760706165998259795, −6.06336505860867497931295167652, −5.39832916075589557651562443045, −4.25652550087061325565099291555, −3.83016681977890720995791511952, −2.82202567463378342340885637746, −1.76606932584645474735627190650, −0.57799063690462197429439185325, 0.57799063690462197429439185325, 1.76606932584645474735627190650, 2.82202567463378342340885637746, 3.83016681977890720995791511952, 4.25652550087061325565099291555, 5.39832916075589557651562443045, 6.06336505860867497931295167652, 6.48099804983760706165998259795, 7.30423973325718076760557733814, 7.919407638442506951583073500571

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