Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.707 - 0.707i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.618i·3-s + i·7-s + 0.618·9-s + i·11-s + 0.618·13-s − 17-s + 0.618i·19-s + 0.618·21-s i·27-s − 29-s + 1.61i·31-s + 0.618·33-s − 0.381i·39-s + 41-s + i·43-s + ⋯
L(s)  = 1  − 0.618i·3-s + i·7-s + 0.618·9-s + i·11-s + 0.618·13-s − 17-s + 0.618i·19-s + 0.618·21-s i·27-s − 29-s + 1.61i·31-s + 0.618·33-s − 0.381i·39-s + 41-s + i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.707 - 0.707i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (2751, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 4000,\ (\ :0),\ 0.707 - 0.707i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.279793340\)
\(L(\frac12)\)  \(\approx\)  \(1.279793340\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + 0.618iT - T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 - 0.618T + T^{2} \)
17 \( 1 + T + T^{2} \)
19 \( 1 - 0.618iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 - 1.61iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 - 1.61iT - T^{2} \)
53 \( 1 + 1.61T + T^{2} \)
59 \( 1 + 1.61iT - T^{2} \)
61 \( 1 - 0.618T + T^{2} \)
67 \( 1 + 1.61iT - T^{2} \)
71 \( 1 - iT - T^{2} \)
73 \( 1 + 1.61T + T^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.61T + T^{2} \)
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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

−8.697862580920212417814460473286, −7.911185273167649129787707467997, −7.27773933945870132432609727737, −6.48323206999738062263551776983, −5.97569036803006198942424979041, −4.92229870832455759520692965289, −4.27921128706487759394645745623, −3.16281552439781541146264152255, −2.10026071645969598454327607370, −1.49857196349192244242743947173, 0.74316149484627271561463929145, 2.06816803820930480604423905075, 3.33921066949657398906188169899, 4.02583803413308960917667996309, 4.51178636250351241981734582257, 5.57281628010657758576924499425, 6.30221258020299763965376251192, 7.18401611918434118011068748555, 7.67002450574604537419499466069, 8.798808739723886242674689971012

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