Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.78·3-s + 2.77·5-s − 7-s + 4.75·9-s − 11-s − 13-s − 7.72·15-s − 2.84·17-s − 6.75·19-s + 2.78·21-s + 0.282·23-s + 2.69·25-s − 4.87·27-s + 8.65·29-s − 2.61·31-s + 2.78·33-s − 2.77·35-s − 2.19·37-s + 2.78·39-s + 9.09·41-s + 6.50·43-s + 13.1·45-s + 0.120·47-s + 49-s + 7.92·51-s + 5.22·53-s − 2.77·55-s + ⋯
L(s)  = 1  − 1.60·3-s + 1.24·5-s − 0.377·7-s + 1.58·9-s − 0.301·11-s − 0.277·13-s − 1.99·15-s − 0.690·17-s − 1.55·19-s + 0.607·21-s + 0.0588·23-s + 0.538·25-s − 0.939·27-s + 1.60·29-s − 0.469·31-s + 0.484·33-s − 0.468·35-s − 0.360·37-s + 0.445·39-s + 1.42·41-s + 0.992·43-s + 1.96·45-s + 0.0176·47-s + 0.142·49-s + 1.10·51-s + 0.718·53-s − 0.373·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9705364144$
$L(\frac12)$  $\approx$  $0.9705364144$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + T \)
good3 \( 1 + 2.78T + 3T^{2} \)
5 \( 1 - 2.77T + 5T^{2} \)
17 \( 1 + 2.84T + 17T^{2} \)
19 \( 1 + 6.75T + 19T^{2} \)
23 \( 1 - 0.282T + 23T^{2} \)
29 \( 1 - 8.65T + 29T^{2} \)
31 \( 1 + 2.61T + 31T^{2} \)
37 \( 1 + 2.19T + 37T^{2} \)
41 \( 1 - 9.09T + 41T^{2} \)
43 \( 1 - 6.50T + 43T^{2} \)
47 \( 1 - 0.120T + 47T^{2} \)
53 \( 1 - 5.22T + 53T^{2} \)
59 \( 1 - 4.04T + 59T^{2} \)
61 \( 1 + 9.87T + 61T^{2} \)
67 \( 1 + 2.78T + 67T^{2} \)
71 \( 1 + 9.02T + 71T^{2} \)
73 \( 1 + 9.94T + 73T^{2} \)
79 \( 1 - 10.1T + 79T^{2} \)
83 \( 1 + 8.81T + 83T^{2} \)
89 \( 1 - 17.7T + 89T^{2} \)
97 \( 1 + 8.81T + 97T^{2} \)
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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

−8.623928921211787809002377580346, −7.41590186106720844446242658082, −6.65348052649924981724117563696, −6.10600944334416523853602076440, −5.73059457978916894186375750266, −4.79636746124946649397289352309, −4.25946673115780002233438751846, −2.71720456969166454233443933513, −1.86491000695698988415832241966, −0.60199209395898874579751363465, 0.60199209395898874579751363465, 1.86491000695698988415832241966, 2.71720456969166454233443933513, 4.25946673115780002233438751846, 4.79636746124946649397289352309, 5.73059457978916894186375750266, 6.10600944334416523853602076440, 6.65348052649924981724117563696, 7.41590186106720844446242658082, 8.623928921211787809002377580346

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