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  + 1.62·3-s + 5-s + 1.05·7-s − 0.374·9-s + 1.92·11-s − 5.16·13-s + 1.62·15-s − 0.487·17-s + 3.35·19-s + 1.71·21-s − 8.10·23-s + 25-s − 5.46·27-s − 5.06·29-s − 9.15·31-s + 3.12·33-s + 1.05·35-s − 2.78·37-s − 8.36·39-s − 9.74·41-s + 9.98·43-s − 0.374·45-s − 12.5·47-s − 5.88·49-s − 0.789·51-s − 8.21·53-s + 1.92·55-s + ⋯
L(s)  = 1  + 0.935·3-s + 0.447·5-s + 0.399·7-s − 0.124·9-s + 0.581·11-s − 1.43·13-s + 0.418·15-s − 0.118·17-s + 0.769·19-s + 0.374·21-s − 1.68·23-s + 0.200·25-s − 1.05·27-s − 0.940·29-s − 1.64·31-s + 0.544·33-s + 0.178·35-s − 0.458·37-s − 1.34·39-s − 1.52·41-s + 1.52·43-s − 0.0558·45-s − 1.83·47-s − 0.840·49-s − 0.110·51-s − 1.12·53-s + 0.260·55-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 \( 1 - 1.62T + 3T^{2} \)
7 \( 1 - 1.05T + 7T^{2} \)
11 \( 1 - 1.92T + 11T^{2} \)
13 \( 1 + 5.16T + 13T^{2} \)
17 \( 1 + 0.487T + 17T^{2} \)
19 \( 1 - 3.35T + 19T^{2} \)
23 \( 1 + 8.10T + 23T^{2} \)
29 \( 1 + 5.06T + 29T^{2} \)
31 \( 1 + 9.15T + 31T^{2} \)
37 \( 1 + 2.78T + 37T^{2} \)
41 \( 1 + 9.74T + 41T^{2} \)
43 \( 1 - 9.98T + 43T^{2} \)
47 \( 1 + 12.5T + 47T^{2} \)
53 \( 1 + 8.21T + 53T^{2} \)
59 \( 1 - 6.65T + 59T^{2} \)
61 \( 1 - 7.66T + 61T^{2} \)
67 \( 1 - 13.0T + 67T^{2} \)
71 \( 1 + 15.2T + 71T^{2} \)
73 \( 1 + 5.43T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 - 5.38T + 83T^{2} \)
89 \( 1 + 5.15T + 89T^{2} \)
97 \( 1 + 4.23T + 97T^{2} \)
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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

−7.70367200842431650857473697731, −7.27260384900480107275230090277, −6.31724875950791081948211142343, −5.49183321623260799017248941819, −4.89152612288433592901537684336, −3.84096151373590789246927491420, −3.23996094957161974492260628412, −2.15747130424140685227181067600, −1.77591427713038857411955999872, 0, 1.77591427713038857411955999872, 2.15747130424140685227181067600, 3.23996094957161974492260628412, 3.84096151373590789246927491420, 4.89152612288433592901537684336, 5.49183321623260799017248941819, 6.31724875950791081948211142343, 7.27260384900480107275230090277, 7.70367200842431650857473697731

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