Properties

Degree 2
Conductor $ 2^{2} $
Sign $-1$
Motivic weight 11
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 516·3-s − 1.05e4·5-s + 4.93e4·7-s + 8.91e4·9-s − 3.09e5·11-s − 1.72e6·13-s + 5.43e6·15-s − 2.27e6·17-s + 4.55e6·19-s − 2.54e7·21-s − 7.28e6·23-s + 6.20e7·25-s + 4.54e7·27-s − 6.90e7·29-s − 1.41e8·31-s + 1.59e8·33-s − 5.19e8·35-s + 7.11e8·37-s + 8.89e8·39-s − 1.22e9·41-s − 3.36e7·43-s − 9.38e8·45-s + 1.23e8·47-s + 4.53e8·49-s + 1.17e9·51-s + 1.10e9·53-s + 3.25e9·55-s + ⋯
L(s)  = 1  − 1.22·3-s − 1.50·5-s + 1.10·7-s + 0.503·9-s − 0.579·11-s − 1.28·13-s + 1.84·15-s − 0.389·17-s + 0.421·19-s − 1.35·21-s − 0.235·23-s + 1.27·25-s + 0.609·27-s − 0.625·29-s − 0.889·31-s + 0.710·33-s − 1.67·35-s + 1.68·37-s + 1.57·39-s − 1.65·41-s − 0.0348·43-s − 0.758·45-s + 0.0783·47-s + 0.229·49-s + 0.477·51-s + 0.363·53-s + 0.872·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(11\)
character  :  $\chi_{4} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4,\ (\ :11/2),\ -1)$
$L(6)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{13}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
good3 \( 1 + 172 p T + p^{11} T^{2} \)
5 \( 1 + 2106 p T + p^{11} T^{2} \)
7 \( 1 - 49304 T + p^{11} T^{2} \)
11 \( 1 + 309420 T + p^{11} T^{2} \)
13 \( 1 + 1723594 T + p^{11} T^{2} \)
17 \( 1 + 2279502 T + p^{11} T^{2} \)
19 \( 1 - 4550444 T + p^{11} T^{2} \)
23 \( 1 + 7282872 T + p^{11} T^{2} \)
29 \( 1 + 69040026 T + p^{11} T^{2} \)
31 \( 1 + 141740704 T + p^{11} T^{2} \)
37 \( 1 - 711366974 T + p^{11} T^{2} \)
41 \( 1 + 1225262214 T + p^{11} T^{2} \)
43 \( 1 + 781540 p T + p^{11} T^{2} \)
47 \( 1 - 123214608 T + p^{11} T^{2} \)
53 \( 1 - 1106121582 T + p^{11} T^{2} \)
59 \( 1 + 9062779932 T + p^{11} T^{2} \)
61 \( 1 + 3854150458 T + p^{11} T^{2} \)
67 \( 1 + 15313764676 T + p^{11} T^{2} \)
71 \( 1 - 20619626328 T + p^{11} T^{2} \)
73 \( 1 + 2063718694 T + p^{11} T^{2} \)
79 \( 1 - 13689871472 T + p^{11} T^{2} \)
83 \( 1 - 65570428908 T + p^{11} T^{2} \)
89 \( 1 + 29715508854 T + p^{11} T^{2} \)
97 \( 1 + 23439626206 T + p^{11} T^{2} \)
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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

−22.06109866433068452206172557423, −20.04950916627822221709274531585, −18.20947205296923602811462763920, −16.69000482504312290024239118396, −15.06487205383862800159819581966, −12.08409413443952132970218524204, −11.07944925540005663929620431223, −7.67375813661486717198729788923, −4.85967772325160364098586662825, 0, 4.85967772325160364098586662825, 7.67375813661486717198729788923, 11.07944925540005663929620431223, 12.08409413443952132970218524204, 15.06487205383862800159819581966, 16.69000482504312290024239118396, 18.20947205296923602811462763920, 20.04950916627822221709274531585, 22.06109866433068452206172557423

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