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  + 0.325·3-s + 5-s + 2.31·7-s − 2.89·9-s + 0.0563·11-s − 6.43·13-s + 0.325·15-s + 3.42·17-s − 2.17·19-s + 0.753·21-s − 3.43·23-s + 25-s − 1.91·27-s + 9.17·29-s − 4.04·31-s + 0.0183·33-s + 2.31·35-s − 2.68·37-s − 2.09·39-s + 0.966·41-s + 3.65·43-s − 2.89·45-s + 9.41·47-s − 1.63·49-s + 1.11·51-s − 3.66·53-s + 0.0563·55-s + ⋯
L(s)  = 1  + 0.187·3-s + 0.447·5-s + 0.875·7-s − 0.964·9-s + 0.0169·11-s − 1.78·13-s + 0.0839·15-s + 0.831·17-s − 0.499·19-s + 0.164·21-s − 0.716·23-s + 0.200·25-s − 0.368·27-s + 1.70·29-s − 0.726·31-s + 0.00318·33-s + 0.391·35-s − 0.441·37-s − 0.335·39-s + 0.150·41-s + 0.556·43-s − 0.431·45-s + 1.37·47-s − 0.233·49-s + 0.156·51-s − 0.503·53-s + 0.00759·55-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 \( 1 - 0.325T + 3T^{2} \)
7 \( 1 - 2.31T + 7T^{2} \)
11 \( 1 - 0.0563T + 11T^{2} \)
13 \( 1 + 6.43T + 13T^{2} \)
17 \( 1 - 3.42T + 17T^{2} \)
19 \( 1 + 2.17T + 19T^{2} \)
23 \( 1 + 3.43T + 23T^{2} \)
29 \( 1 - 9.17T + 29T^{2} \)
31 \( 1 + 4.04T + 31T^{2} \)
37 \( 1 + 2.68T + 37T^{2} \)
41 \( 1 - 0.966T + 41T^{2} \)
43 \( 1 - 3.65T + 43T^{2} \)
47 \( 1 - 9.41T + 47T^{2} \)
53 \( 1 + 3.66T + 53T^{2} \)
59 \( 1 + 8.00T + 59T^{2} \)
61 \( 1 - 0.207T + 61T^{2} \)
67 \( 1 + 3.39T + 67T^{2} \)
71 \( 1 + 5.80T + 71T^{2} \)
73 \( 1 + 6.19T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 + 6.06T + 83T^{2} \)
89 \( 1 + 5.64T + 89T^{2} \)
97 \( 1 + 7.46T + 97T^{2} \)
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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

−7.74140330070220516367386786239, −7.18682459027707898104062683070, −6.18047530042526681993947047018, −5.52824893647773334966296890001, −4.89734973992545863367322745771, −4.18812772813338949760737273047, −2.94140182627710275229707339751, −2.43536233040450065063624322559, −1.45791412774876054039557994235, 0, 1.45791412774876054039557994235, 2.43536233040450065063624322559, 2.94140182627710275229707339751, 4.18812772813338949760737273047, 4.89734973992545863367322745771, 5.52824893647773334966296890001, 6.18047530042526681993947047018, 7.18682459027707898104062683070, 7.74140330070220516367386786239

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