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 + 2.09·5-s − 1.11·7-s + 9-s − 3.05·11-s − 6.80·13-s − 2.09·15-s + 6.88·17-s − 6.06·19-s + 1.11·21-s − 0.508·23-s − 0.623·25-s − 27-s − 4.52·29-s + 1.62·31-s + 3.05·33-s − 2.34·35-s + 4.52·37-s + 6.80·39-s + 3.40·41-s − 7.55·43-s + 2.09·45-s + 12.9·47-s − 5.74·49-s − 6.88·51-s + 5.82·53-s − 6.39·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.935·5-s − 0.423·7-s + 0.333·9-s − 0.922·11-s − 1.88·13-s − 0.540·15-s + 1.66·17-s − 1.39·19-s + 0.244·21-s − 0.105·23-s − 0.124·25-s − 0.192·27-s − 0.840·29-s + 0.291·31-s + 0.532·33-s − 0.396·35-s + 0.744·37-s + 1.09·39-s + 0.531·41-s − 1.15·43-s + 0.311·45-s + 1.89·47-s − 0.820·49-s − 0.964·51-s + 0.800·53-s − 0.862·55-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  =  $(2,\ 6036,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.208371921$
$L(\frac12)$  $\approx$  $1.208371921$
$L(\frac{3}{2})$   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 \( 1 - 2.09T + 5T^{2} \)
7 \( 1 + 1.11T + 7T^{2} \)
11 \( 1 + 3.05T + 11T^{2} \)
13 \( 1 + 6.80T + 13T^{2} \)
17 \( 1 - 6.88T + 17T^{2} \)
19 \( 1 + 6.06T + 19T^{2} \)
23 \( 1 + 0.508T + 23T^{2} \)
29 \( 1 + 4.52T + 29T^{2} \)
31 \( 1 - 1.62T + 31T^{2} \)
37 \( 1 - 4.52T + 37T^{2} \)
41 \( 1 - 3.40T + 41T^{2} \)
43 \( 1 + 7.55T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 - 5.82T + 53T^{2} \)
59 \( 1 - 7.98T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 7.90T + 67T^{2} \)
71 \( 1 - 3.42T + 71T^{2} \)
73 \( 1 - 14.4T + 73T^{2} \)
79 \( 1 + 4.06T + 79T^{2} \)
83 \( 1 + 4.08T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 14.8T + 97T^{2} \)
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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

−7.77784083342299600113467106272, −7.48247846682177705788562298545, −6.51050001114915027187257753151, −5.87852217667309090571053042663, −5.29096764189060074805078010287, −4.71284021966117998849612439353, −3.63089395673223638479596461208, −2.54190025187079191516241932950, −2.02762157373263447637665431106, −0.56562823562890893731280143581, 0.56562823562890893731280143581, 2.02762157373263447637665431106, 2.54190025187079191516241932950, 3.63089395673223638479596461208, 4.71284021966117998849612439353, 5.29096764189060074805078010287, 5.87852217667309090571053042663, 6.51050001114915027187257753151, 7.48247846682177705788562298545, 7.77784083342299600113467106272

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