Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 23 \cdot 29 $
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 − 0.812·5-s − 4.94·7-s + 9-s + 1.85·11-s − 6.88·13-s − 0.812·15-s − 7.62·17-s + 0.765·19-s − 4.94·21-s + 23-s − 4.34·25-s + 27-s − 29-s − 7.16·31-s + 1.85·33-s + 4.01·35-s − 6.94·37-s − 6.88·39-s − 2.06·41-s + 12.9·43-s − 0.812·45-s − 3.03·47-s + 17.4·49-s − 7.62·51-s + 13.0·53-s − 1.50·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.363·5-s − 1.86·7-s + 0.333·9-s + 0.559·11-s − 1.90·13-s − 0.209·15-s − 1.84·17-s + 0.175·19-s − 1.07·21-s + 0.208·23-s − 0.868·25-s + 0.192·27-s − 0.185·29-s − 1.28·31-s + 0.323·33-s + 0.678·35-s − 1.14·37-s − 1.10·39-s − 0.322·41-s + 1.97·43-s − 0.121·45-s − 0.443·47-s + 2.49·49-s − 1.06·51-s + 1.79·53-s − 0.203·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{8004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.6977347338$
$L(\frac12)$  $\approx$  $0.6977347338$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
good5 \( 1 + 0.812T + 5T^{2} \)
7 \( 1 + 4.94T + 7T^{2} \)
11 \( 1 - 1.85T + 11T^{2} \)
13 \( 1 + 6.88T + 13T^{2} \)
17 \( 1 + 7.62T + 17T^{2} \)
19 \( 1 - 0.765T + 19T^{2} \)
31 \( 1 + 7.16T + 31T^{2} \)
37 \( 1 + 6.94T + 37T^{2} \)
41 \( 1 + 2.06T + 41T^{2} \)
43 \( 1 - 12.9T + 43T^{2} \)
47 \( 1 + 3.03T + 47T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 - 6.09T + 59T^{2} \)
61 \( 1 + 3.49T + 61T^{2} \)
67 \( 1 + 9.90T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 + 3.62T + 73T^{2} \)
79 \( 1 - 15.4T + 79T^{2} \)
83 \( 1 - 0.720T + 83T^{2} \)
89 \( 1 - 4.99T + 89T^{2} \)
97 \( 1 + 5.22T + 97T^{2} \)
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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

−7.60929726419638821332242434053, −7.10347375961826564706446928957, −6.71260103978835685221179108793, −5.86417379135726990973995045846, −4.94566828595683956615819503703, −4.02212550830512123224955581619, −3.60130225143883864581617015602, −2.60109442878555114956150671420, −2.13641410253276904562852837446, −0.36840260269670983551651814595, 0.36840260269670983551651814595, 2.13641410253276904562852837446, 2.60109442878555114956150671420, 3.60130225143883864581617015602, 4.02212550830512123224955581619, 4.94566828595683956615819503703, 5.86417379135726990973995045846, 6.71260103978835685221179108793, 7.10347375961826564706446928957, 7.60929726419638821332242434053

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