Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2.01·5-s − 2.53·7-s + 9-s + 4.24·11-s − 2.95·13-s − 2.01·15-s + 7.86·17-s − 7.02·19-s − 2.53·21-s − 2.38·23-s − 0.921·25-s + 27-s − 0.629·29-s + 4.91·31-s + 4.24·33-s + 5.11·35-s + 1.38·37-s − 2.95·39-s + 4.74·41-s − 1.92·43-s − 2.01·45-s + 4.57·47-s − 0.586·49-s + 7.86·51-s − 5.90·53-s − 8.57·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.903·5-s − 0.957·7-s + 0.333·9-s + 1.27·11-s − 0.818·13-s − 0.521·15-s + 1.90·17-s − 1.61·19-s − 0.552·21-s − 0.496·23-s − 0.184·25-s + 0.192·27-s − 0.116·29-s + 0.883·31-s + 0.738·33-s + 0.864·35-s + 0.227·37-s − 0.472·39-s + 0.740·41-s − 0.292·43-s − 0.301·45-s + 0.667·47-s − 0.0837·49-s + 1.10·51-s − 0.810·53-s − 1.15·55-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  =  1
Selberg data  =  $(2,\ 6036,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + 2.01T + 5T^{2} \)
7 \( 1 + 2.53T + 7T^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + 2.95T + 13T^{2} \)
17 \( 1 - 7.86T + 17T^{2} \)
19 \( 1 + 7.02T + 19T^{2} \)
23 \( 1 + 2.38T + 23T^{2} \)
29 \( 1 + 0.629T + 29T^{2} \)
31 \( 1 - 4.91T + 31T^{2} \)
37 \( 1 - 1.38T + 37T^{2} \)
41 \( 1 - 4.74T + 41T^{2} \)
43 \( 1 + 1.92T + 43T^{2} \)
47 \( 1 - 4.57T + 47T^{2} \)
53 \( 1 + 5.90T + 53T^{2} \)
59 \( 1 - 7.09T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 5.20T + 67T^{2} \)
71 \( 1 + 15.1T + 71T^{2} \)
73 \( 1 - 5.38T + 73T^{2} \)
79 \( 1 - 8.24T + 79T^{2} \)
83 \( 1 + 3.55T + 83T^{2} \)
89 \( 1 + 6.50T + 89T^{2} \)
97 \( 1 + 17.5T + 97T^{2} \)
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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

−7.84968692252616779737942319246, −7.05764734746185259340095819694, −6.43126188909939301024839959058, −5.72093543724890486620360983648, −4.49510547536301526853655144324, −3.95672299548191693129077558326, −3.33315044520020396324100100670, −2.50720200210585852112573438092, −1.29125001879399607322396531797, 0, 1.29125001879399607322396531797, 2.50720200210585852112573438092, 3.33315044520020396324100100670, 3.95672299548191693129077558326, 4.49510547536301526853655144324, 5.72093543724890486620360983648, 6.43126188909939301024839959058, 7.05764734746185259340095819694, 7.84968692252616779737942319246

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