Properties

Degree 2
Conductor $ 2^{2} $
Sign $1$
Motivic weight 17
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.15e4·3-s + 9.70e5·5-s + 1.64e6·7-s + 3.33e8·9-s − 7.82e7·11-s + 2.36e9·13-s − 2.08e10·15-s − 1.04e10·17-s + 1.03e11·19-s − 3.53e10·21-s + 3.30e11·23-s + 1.79e11·25-s − 4.40e12·27-s + 3.69e12·29-s + 2.53e12·31-s + 1.68e12·33-s + 1.59e12·35-s + 1.96e13·37-s − 5.08e13·39-s − 3.32e13·41-s − 8.49e13·43-s + 3.24e14·45-s + 2.18e12·47-s − 2.29e14·49-s + 2.25e14·51-s − 1.81e13·53-s − 7.59e13·55-s + ⋯
L(s)  = 1  − 1.89·3-s + 1.11·5-s + 0.107·7-s + 2.58·9-s − 0.110·11-s + 0.803·13-s − 2.10·15-s − 0.364·17-s + 1.40·19-s − 0.203·21-s + 0.881·23-s + 0.235·25-s − 3.00·27-s + 1.37·29-s + 0.533·31-s + 0.208·33-s + 0.119·35-s + 0.919·37-s − 1.52·39-s − 0.650·41-s − 1.10·43-s + 2.87·45-s + 0.0133·47-s − 0.988·49-s + 0.691·51-s − 0.0401·53-s − 0.122·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(17\)
character  :  $\chi_{4} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4,\ (\ :17/2),\ 1)$
$L(9)$  $\approx$  $1.13846$
$L(\frac12)$  $\approx$  $1.13846$
$L(\frac{19}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p(T)\) is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 2.15e4T + 1.29e8T^{2} \)
5 \( 1 - 9.70e5T + 7.62e11T^{2} \)
7 \( 1 - 1.64e6T + 2.32e14T^{2} \)
11 \( 1 + 7.82e7T + 5.05e17T^{2} \)
13 \( 1 - 2.36e9T + 8.65e18T^{2} \)
17 \( 1 + 1.04e10T + 8.27e20T^{2} \)
19 \( 1 - 1.03e11T + 5.48e21T^{2} \)
23 \( 1 - 3.30e11T + 1.41e23T^{2} \)
29 \( 1 - 3.69e12T + 7.25e24T^{2} \)
31 \( 1 - 2.53e12T + 2.25e25T^{2} \)
37 \( 1 - 1.96e13T + 4.56e26T^{2} \)
41 \( 1 + 3.32e13T + 2.61e27T^{2} \)
43 \( 1 + 8.49e13T + 5.87e27T^{2} \)
47 \( 1 - 2.18e12T + 2.66e28T^{2} \)
53 \( 1 + 1.81e13T + 2.05e29T^{2} \)
59 \( 1 - 1.24e15T + 1.27e30T^{2} \)
61 \( 1 - 2.09e15T + 2.24e30T^{2} \)
67 \( 1 + 2.53e15T + 1.10e31T^{2} \)
71 \( 1 - 8.39e15T + 2.96e31T^{2} \)
73 \( 1 + 6.50e15T + 4.74e31T^{2} \)
79 \( 1 + 1.41e16T + 1.81e32T^{2} \)
83 \( 1 - 1.76e16T + 4.21e32T^{2} \)
89 \( 1 - 2.17e16T + 1.37e33T^{2} \)
97 \( 1 + 8.31e16T + 5.95e33T^{2} \)
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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

−21.17200737763757841361731633284, −18.27783764260195604388851924085, −17.42272531046447011513111920059, −16.04523214199944693054209519596, −13.28890716299919749587649571602, −11.52409998933883939093132771380, −10.04523180569335592908663683829, −6.48035812582763516333469305742, −5.16076977806956282797043243787, −1.11495662939605257989208254741, 1.11495662939605257989208254741, 5.16076977806956282797043243787, 6.48035812582763516333469305742, 10.04523180569335592908663683829, 11.52409998933883939093132771380, 13.28890716299919749587649571602, 16.04523214199944693054209519596, 17.42272531046447011513111920059, 18.27783764260195604388851924085, 21.17200737763757841361731633284

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