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 + 3.34·5-s + 2.75·7-s + 9-s + 2.38·11-s + 3.79·13-s − 3.34·15-s + 4.35·17-s − 8.55·19-s − 2.75·21-s − 8.57·23-s + 6.16·25-s − 27-s + 5.46·29-s − 9.44·31-s − 2.38·33-s + 9.19·35-s + 11.3·37-s − 3.79·39-s + 0.318·41-s + 6.22·43-s + 3.34·45-s + 10.3·47-s + 0.569·49-s − 4.35·51-s + 8.92·53-s + 7.96·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.49·5-s + 1.03·7-s + 0.333·9-s + 0.718·11-s + 1.05·13-s − 0.862·15-s + 1.05·17-s − 1.96·19-s − 0.600·21-s − 1.78·23-s + 1.23·25-s − 0.192·27-s + 1.01·29-s − 1.69·31-s − 0.414·33-s + 1.55·35-s + 1.86·37-s − 0.608·39-s + 0.0497·41-s + 0.949·43-s + 0.498·45-s + 1.51·47-s + 0.0813·49-s − 0.610·51-s + 1.22·53-s + 1.07·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.914606384$
$L(\frac12)$  $\approx$  $2.914606384$
$L(\frac{3}{2})$   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 \( 1 - 3.34T + 5T^{2} \)
7 \( 1 - 2.75T + 7T^{2} \)
11 \( 1 - 2.38T + 11T^{2} \)
13 \( 1 - 3.79T + 13T^{2} \)
17 \( 1 - 4.35T + 17T^{2} \)
19 \( 1 + 8.55T + 19T^{2} \)
23 \( 1 + 8.57T + 23T^{2} \)
29 \( 1 - 5.46T + 29T^{2} \)
31 \( 1 + 9.44T + 31T^{2} \)
37 \( 1 - 11.3T + 37T^{2} \)
41 \( 1 - 0.318T + 41T^{2} \)
43 \( 1 - 6.22T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 - 8.92T + 53T^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 - 2.18T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 + 9.42T + 71T^{2} \)
73 \( 1 + 15.9T + 73T^{2} \)
79 \( 1 + 0.894T + 79T^{2} \)
83 \( 1 + 14.2T + 83T^{2} \)
89 \( 1 + 13.4T + 89T^{2} \)
97 \( 1 - 12.1T + 97T^{2} \)
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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

−8.229071249266493996159746721088, −7.26092873689132861562819712308, −6.39801105296166995691814705176, −5.74687445109503843728319763879, −5.66207062308814630124387712648, −4.33715390769587114113812842983, −3.98960485961706288458505807802, −2.42544203466916456695080521862, −1.78024007900327453637788958751, −1.00614588856557401035785748015, 1.00614588856557401035785748015, 1.78024007900327453637788958751, 2.42544203466916456695080521862, 3.98960485961706288458505807802, 4.33715390769587114113812842983, 5.66207062308814630124387712648, 5.74687445109503843728319763879, 6.39801105296166995691814705176, 7.26092873689132861562819712308, 8.229071249266493996159746721088

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