Properties

Degree 2
Conductor $ 2^{3} \cdot 5 \cdot 151 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.75·3-s + 5-s + 1.53·7-s + 0.0913·9-s − 5.86·11-s + 0.896·13-s − 1.75·15-s − 5.84·17-s + 3.50·19-s − 2.69·21-s + 2.74·23-s + 25-s + 5.11·27-s + 2.53·29-s + 5.38·31-s + 10.3·33-s + 1.53·35-s + 1.58·37-s − 1.57·39-s + 1.42·41-s + 8.60·43-s + 0.0913·45-s + 1.37·47-s − 4.65·49-s + 10.2·51-s − 10.7·53-s − 5.86·55-s + ⋯
L(s)  = 1  − 1.01·3-s + 0.447·5-s + 0.578·7-s + 0.0304·9-s − 1.76·11-s + 0.248·13-s − 0.453·15-s − 1.41·17-s + 0.804·19-s − 0.587·21-s + 0.571·23-s + 0.200·25-s + 0.984·27-s + 0.469·29-s + 0.966·31-s + 1.79·33-s + 0.258·35-s + 0.260·37-s − 0.252·39-s + 0.222·41-s + 1.31·43-s + 0.0136·45-s + 0.199·47-s − 0.664·49-s + 1.44·51-s − 1.47·53-s − 0.790·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6040\)    =    \(2^{3} \cdot 5 \cdot 151\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6040} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6040,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5,\;151\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;5,\;151\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 - T \)
151 \( 1 + T \)
good3 \( 1 + 1.75T + 3T^{2} \)
7 \( 1 - 1.53T + 7T^{2} \)
11 \( 1 + 5.86T + 11T^{2} \)
13 \( 1 - 0.896T + 13T^{2} \)
17 \( 1 + 5.84T + 17T^{2} \)
19 \( 1 - 3.50T + 19T^{2} \)
23 \( 1 - 2.74T + 23T^{2} \)
29 \( 1 - 2.53T + 29T^{2} \)
31 \( 1 - 5.38T + 31T^{2} \)
37 \( 1 - 1.58T + 37T^{2} \)
41 \( 1 - 1.42T + 41T^{2} \)
43 \( 1 - 8.60T + 43T^{2} \)
47 \( 1 - 1.37T + 47T^{2} \)
53 \( 1 + 10.7T + 53T^{2} \)
59 \( 1 + 0.677T + 59T^{2} \)
61 \( 1 - 8.86T + 61T^{2} \)
67 \( 1 + 9.39T + 67T^{2} \)
71 \( 1 + 4.08T + 71T^{2} \)
73 \( 1 + 4.48T + 73T^{2} \)
79 \( 1 - 6.93T + 79T^{2} \)
83 \( 1 + 5.69T + 83T^{2} \)
89 \( 1 + 3.48T + 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−7.72658892127455723797462730449, −6.90010849808164257648585863730, −6.16311759447646223802947221947, −5.54975049636331727317113496457, −4.92386575222181122895694284986, −4.46085672658379480264785109942, −2.97520130322760833077331801771, −2.40034872528875233770648228034, −1.14494982582733615719429592562, 0, 1.14494982582733615719429592562, 2.40034872528875233770648228034, 2.97520130322760833077331801771, 4.46085672658379480264785109942, 4.92386575222181122895694284986, 5.54975049636331727317113496457, 6.16311759447646223802947221947, 6.90010849808164257648585863730, 7.72658892127455723797462730449

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