Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 251 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.38·5-s + 1.87·7-s − 0.536·11-s − 3.43·13-s + 5.29·17-s + 2.51·19-s − 1.08·23-s − 3.09·25-s − 1.95·29-s − 10.5·31-s + 2.58·35-s − 6.13·37-s − 6.36·41-s − 0.652·43-s − 10.2·47-s − 3.50·49-s + 1.31·53-s − 0.741·55-s + 13.8·59-s − 8.27·61-s − 4.74·65-s − 12.4·67-s + 8.24·71-s + 7.43·73-s − 1.00·77-s − 7.33·79-s − 7.83·83-s + ⋯
L(s)  = 1  + 0.617·5-s + 0.706·7-s − 0.161·11-s − 0.952·13-s + 1.28·17-s + 0.577·19-s − 0.225·23-s − 0.618·25-s − 0.362·29-s − 1.89·31-s + 0.436·35-s − 1.00·37-s − 0.993·41-s − 0.0995·43-s − 1.50·47-s − 0.500·49-s + 0.181·53-s − 0.0999·55-s + 1.79·59-s − 1.05·61-s − 0.588·65-s − 1.52·67-s + 0.977·71-s + 0.869·73-s − 0.114·77-s − 0.825·79-s − 0.859·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(9036\)    =    \(2^{2} \cdot 3^{2} \cdot 251\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{9036} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 9036,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;251\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;251\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
251 \( 1 + T \)
good5 \( 1 - 1.38T + 5T^{2} \)
7 \( 1 - 1.87T + 7T^{2} \)
11 \( 1 + 0.536T + 11T^{2} \)
13 \( 1 + 3.43T + 13T^{2} \)
17 \( 1 - 5.29T + 17T^{2} \)
19 \( 1 - 2.51T + 19T^{2} \)
23 \( 1 + 1.08T + 23T^{2} \)
29 \( 1 + 1.95T + 29T^{2} \)
31 \( 1 + 10.5T + 31T^{2} \)
37 \( 1 + 6.13T + 37T^{2} \)
41 \( 1 + 6.36T + 41T^{2} \)
43 \( 1 + 0.652T + 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 1.31T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 8.27T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 - 8.24T + 71T^{2} \)
73 \( 1 - 7.43T + 73T^{2} \)
79 \( 1 + 7.33T + 79T^{2} \)
83 \( 1 + 7.83T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−7.41860368472373572687062926914, −6.85192314469154855627497351145, −5.76528409872871698318230309132, −5.38292898018971543739559252621, −4.82257236415598825945705702052, −3.76152368637042720720801424454, −3.07443237000859342599757915885, −1.99344737750919091937341510044, −1.48934659998638266436912219604, 0, 1.48934659998638266436912219604, 1.99344737750919091937341510044, 3.07443237000859342599757915885, 3.76152368637042720720801424454, 4.82257236415598825945705702052, 5.38292898018971543739559252621, 5.76528409872871698318230309132, 6.85192314469154855627497351145, 7.41860368472373572687062926914

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