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 + 4.08·5-s + 2.57·7-s + 9-s + 4.61·11-s − 4.79·13-s + 4.08·15-s − 1.83·17-s + 4.89·19-s + 2.57·21-s + 23-s + 11.7·25-s + 27-s − 29-s + 8.03·31-s + 4.61·33-s + 10.5·35-s − 8.43·37-s − 4.79·39-s + 8.64·41-s − 7.46·43-s + 4.08·45-s − 5.52·47-s − 0.366·49-s − 1.83·51-s − 7.81·53-s + 18.8·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.82·5-s + 0.973·7-s + 0.333·9-s + 1.39·11-s − 1.33·13-s + 1.05·15-s − 0.444·17-s + 1.12·19-s + 0.562·21-s + 0.208·23-s + 2.34·25-s + 0.192·27-s − 0.185·29-s + 1.44·31-s + 0.802·33-s + 1.78·35-s − 1.38·37-s − 0.768·39-s + 1.35·41-s − 1.13·43-s + 0.609·45-s − 0.806·47-s − 0.0523·49-s − 0.256·51-s − 1.07·53-s + 2.54·55-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  =  $(2,\ 8004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.814340552$
$L(\frac12)$  $\approx$  $4.814340552$
$L(\frac{3}{2})$   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 \( 1 - 4.08T + 5T^{2} \)
7 \( 1 - 2.57T + 7T^{2} \)
11 \( 1 - 4.61T + 11T^{2} \)
13 \( 1 + 4.79T + 13T^{2} \)
17 \( 1 + 1.83T + 17T^{2} \)
19 \( 1 - 4.89T + 19T^{2} \)
31 \( 1 - 8.03T + 31T^{2} \)
37 \( 1 + 8.43T + 37T^{2} \)
41 \( 1 - 8.64T + 41T^{2} \)
43 \( 1 + 7.46T + 43T^{2} \)
47 \( 1 + 5.52T + 47T^{2} \)
53 \( 1 + 7.81T + 53T^{2} \)
59 \( 1 - 7.57T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 - 8.30T + 67T^{2} \)
71 \( 1 + 0.0870T + 71T^{2} \)
73 \( 1 - 9.17T + 73T^{2} \)
79 \( 1 + 8.38T + 79T^{2} \)
83 \( 1 + 9.19T + 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 - 6.00T + 97T^{2} \)
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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

−7.87905067566021583287326284538, −6.99506400964558501249850294469, −6.57864900416905891940992811982, −5.71075278389378831813693057728, −4.96381560764056595775648459417, −4.54709495802633873469249965513, −3.31314168616254975029579341651, −2.50028287319415765291363856510, −1.77301526862498186376390551942, −1.20269040816193260036434779832, 1.20269040816193260036434779832, 1.77301526862498186376390551942, 2.50028287319415765291363856510, 3.31314168616254975029579341651, 4.54709495802633873469249965513, 4.96381560764056595775648459417, 5.71075278389378831813693057728, 6.57864900416905891940992811982, 6.99506400964558501249850294469, 7.87905067566021583287326284538

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