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  + 1.56e4·3-s − 3.66e5·5-s + 2.37e7·7-s + 1.15e8·9-s + 1.33e9·11-s − 3.68e9·13-s − 5.73e9·15-s − 1.70e10·17-s − 2.59e9·19-s + 3.70e11·21-s − 1.96e11·23-s − 6.28e11·25-s − 2.15e11·27-s − 1.36e12·29-s − 2.25e12·31-s + 2.09e13·33-s − 8.69e12·35-s + 1.28e12·37-s − 5.75e13·39-s + 3.74e13·41-s − 2.61e13·43-s − 4.23e13·45-s + 1.94e14·47-s + 3.29e14·49-s − 2.65e14·51-s − 4.69e14·53-s − 4.90e14·55-s + ⋯
L(s)  = 1  + 1.37·3-s − 0.419·5-s + 1.55·7-s + 0.893·9-s + 1.88·11-s − 1.25·13-s − 0.577·15-s − 0.591·17-s − 0.0350·19-s + 2.13·21-s − 0.522·23-s − 0.823·25-s − 0.146·27-s − 0.505·29-s − 0.475·31-s + 2.58·33-s − 0.652·35-s + 0.0602·37-s − 1.72·39-s + 0.732·41-s − 0.341·43-s − 0.375·45-s + 1.19·47-s + 1.41·49-s − 0.813·51-s − 1.03·53-s − 0.790·55-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  =  $(2,\ 4,\ (\ :17/2),\ 1)$
$L(9)$  $\approx$  $2.61817$
$L(\frac12)$  $\approx$  $2.61817$
$L(\frac{19}{2})$   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 \( 1 - 1.56e4T + 1.29e8T^{2} \)
5 \( 1 + 3.66e5T + 7.62e11T^{2} \)
7 \( 1 - 2.37e7T + 2.32e14T^{2} \)
11 \( 1 - 1.33e9T + 5.05e17T^{2} \)
13 \( 1 + 3.68e9T + 8.65e18T^{2} \)
17 \( 1 + 1.70e10T + 8.27e20T^{2} \)
19 \( 1 + 2.59e9T + 5.48e21T^{2} \)
23 \( 1 + 1.96e11T + 1.41e23T^{2} \)
29 \( 1 + 1.36e12T + 7.25e24T^{2} \)
31 \( 1 + 2.25e12T + 2.25e25T^{2} \)
37 \( 1 - 1.28e12T + 4.56e26T^{2} \)
41 \( 1 - 3.74e13T + 2.61e27T^{2} \)
43 \( 1 + 2.61e13T + 5.87e27T^{2} \)
47 \( 1 - 1.94e14T + 2.66e28T^{2} \)
53 \( 1 + 4.69e14T + 2.05e29T^{2} \)
59 \( 1 - 1.58e15T + 1.27e30T^{2} \)
61 \( 1 + 2.16e15T + 2.24e30T^{2} \)
67 \( 1 - 1.54e15T + 1.10e31T^{2} \)
71 \( 1 + 4.51e15T + 2.96e31T^{2} \)
73 \( 1 + 6.24e15T + 4.74e31T^{2} \)
79 \( 1 + 7.95e14T + 1.81e32T^{2} \)
83 \( 1 - 2.62e16T + 4.21e32T^{2} \)
89 \( 1 - 3.01e16T + 1.37e33T^{2} \)
97 \( 1 - 3.38e16T + 5.95e33T^{2} \)
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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

−20.32924792156030931729415985185, −19.42914521055859307916552765402, −17.38258814186511238878605024469, −14.90895416535699676873902099127, −14.18912952905206708297251654682, −11.71454380078477834885769011083, −9.099802903730324013820154414837, −7.66199268442984915857746437293, −4.14802990103025714880467381097, −1.90081812298763992872817716425, 1.90081812298763992872817716425, 4.14802990103025714880467381097, 7.66199268442984915857746437293, 9.099802903730324013820154414837, 11.71454380078477834885769011083, 14.18912952905206708297251654682, 14.90895416535699676873902099127, 17.38258814186511238878605024469, 19.42914521055859307916552765402, 20.32924792156030931729415985185

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