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  − 1.40·3-s + 5-s + 3.27·7-s − 1.03·9-s + 3.84·11-s − 5.20·13-s − 1.40·15-s + 0.810·17-s − 4.07·19-s − 4.59·21-s + 6.38·23-s + 25-s + 5.65·27-s − 7.98·29-s − 4.89·31-s − 5.39·33-s + 3.27·35-s − 7.95·37-s + 7.29·39-s + 10.5·41-s − 7.64·43-s − 1.03·45-s − 6.83·47-s + 3.70·49-s − 1.13·51-s + 1.56·53-s + 3.84·55-s + ⋯
L(s)  = 1  − 0.810·3-s + 0.447·5-s + 1.23·7-s − 0.343·9-s + 1.15·11-s − 1.44·13-s − 0.362·15-s + 0.196·17-s − 0.935·19-s − 1.00·21-s + 1.33·23-s + 0.200·25-s + 1.08·27-s − 1.48·29-s − 0.879·31-s − 0.939·33-s + 0.553·35-s − 1.30·37-s + 1.16·39-s + 1.65·41-s − 1.16·43-s − 0.153·45-s − 0.996·47-s + 0.529·49-s − 0.159·51-s + 0.215·53-s + 0.518·55-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 \( 1 + 1.40T + 3T^{2} \)
7 \( 1 - 3.27T + 7T^{2} \)
11 \( 1 - 3.84T + 11T^{2} \)
13 \( 1 + 5.20T + 13T^{2} \)
17 \( 1 - 0.810T + 17T^{2} \)
19 \( 1 + 4.07T + 19T^{2} \)
23 \( 1 - 6.38T + 23T^{2} \)
29 \( 1 + 7.98T + 29T^{2} \)
31 \( 1 + 4.89T + 31T^{2} \)
37 \( 1 + 7.95T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 7.64T + 43T^{2} \)
47 \( 1 + 6.83T + 47T^{2} \)
53 \( 1 - 1.56T + 53T^{2} \)
59 \( 1 - 10.0T + 59T^{2} \)
61 \( 1 + 7.68T + 61T^{2} \)
67 \( 1 + 14.1T + 67T^{2} \)
71 \( 1 + 10.6T + 71T^{2} \)
73 \( 1 + 8.04T + 73T^{2} \)
79 \( 1 + 5.13T + 79T^{2} \)
83 \( 1 - 0.0490T + 83T^{2} \)
89 \( 1 - 7.75T + 89T^{2} \)
97 \( 1 - 12.6T + 97T^{2} \)
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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

−7.48949316140432001067272907691, −7.07286072358358605769535178921, −6.19650326569722856258621416994, −5.50321141621896381028435706043, −4.93565848074969366661441367961, −4.34627312984043954508529326164, −3.19732019457185512491033874461, −2.09415264406655240356120267044, −1.37667688092273586087692392041, 0, 1.37667688092273586087692392041, 2.09415264406655240356120267044, 3.19732019457185512491033874461, 4.34627312984043954508529326164, 4.93565848074969366661441367961, 5.50321141621896381028435706043, 6.19650326569722856258621416994, 7.07286072358358605769535178921, 7.48949316140432001067272907691

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