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 + 0.897·5-s + 3.81·7-s + 9-s + 5.23·11-s − 3.43·13-s + 0.897·15-s + 1.92·17-s − 7.80·19-s + 3.81·21-s + 23-s − 4.19·25-s + 27-s − 29-s + 3.73·31-s + 5.23·33-s + 3.42·35-s + 0.939·37-s − 3.43·39-s − 5.90·41-s − 2.82·43-s + 0.897·45-s + 13.3·47-s + 7.55·49-s + 1.92·51-s + 12.4·53-s + 4.69·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.401·5-s + 1.44·7-s + 0.333·9-s + 1.57·11-s − 0.952·13-s + 0.231·15-s + 0.466·17-s − 1.79·19-s + 0.832·21-s + 0.208·23-s − 0.838·25-s + 0.192·27-s − 0.185·29-s + 0.670·31-s + 0.911·33-s + 0.578·35-s + 0.154·37-s − 0.549·39-s − 0.922·41-s − 0.430·43-s + 0.133·45-s + 1.94·47-s + 1.07·49-s + 0.269·51-s + 1.70·53-s + 0.633·55-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  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.707053854$
$L(\frac12)$  $\approx$  $3.707053854$
$L(\frac{3}{2})$   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 \( 1 - 0.897T + 5T^{2} \)
7 \( 1 - 3.81T + 7T^{2} \)
11 \( 1 - 5.23T + 11T^{2} \)
13 \( 1 + 3.43T + 13T^{2} \)
17 \( 1 - 1.92T + 17T^{2} \)
19 \( 1 + 7.80T + 19T^{2} \)
31 \( 1 - 3.73T + 31T^{2} \)
37 \( 1 - 0.939T + 37T^{2} \)
41 \( 1 + 5.90T + 41T^{2} \)
43 \( 1 + 2.82T + 43T^{2} \)
47 \( 1 - 13.3T + 47T^{2} \)
53 \( 1 - 12.4T + 53T^{2} \)
59 \( 1 + 3.06T + 59T^{2} \)
61 \( 1 - 9.88T + 61T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 9.23T + 73T^{2} \)
79 \( 1 - 2.22T + 79T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 + 13.2T + 89T^{2} \)
97 \( 1 + 7.70T + 97T^{2} \)
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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

−7.891177975676756911864564926743, −7.19778850161229777576235795388, −6.55761048051466976191660130171, −5.73735108632454518505923999571, −4.91310531708633099265384575742, −4.24078895374149454146625363056, −3.69838342108684601722086285385, −2.32471874204330048893183579576, −1.96893521634020086278555648703, −0.976245152710285356952472904668, 0.976245152710285356952472904668, 1.96893521634020086278555648703, 2.32471874204330048893183579576, 3.69838342108684601722086285385, 4.24078895374149454146625363056, 4.91310531708633099265384575742, 5.73735108632454518505923999571, 6.55761048051466976191660130171, 7.19778850161229777576235795388, 7.891177975676756911864564926743

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