Properties

Degree 2
Conductor $ 2^{3} \cdot 17 \cdot 59 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.61·3-s − 1.23·5-s + 4.85·7-s − 0.394·9-s − 1.78·11-s − 5.69·13-s + 1.99·15-s − 17-s + 2.51·19-s − 7.83·21-s + 6.91·23-s − 3.47·25-s + 5.47·27-s − 7.19·29-s + 5.87·31-s + 2.87·33-s − 5.98·35-s + 3.49·37-s + 9.19·39-s − 1.25·41-s + 1.45·43-s + 0.486·45-s + 2.43·47-s + 16.5·49-s + 1.61·51-s + 2.66·53-s + 2.19·55-s + ⋯
L(s)  = 1  − 0.932·3-s − 0.552·5-s + 1.83·7-s − 0.131·9-s − 0.537·11-s − 1.57·13-s + 0.514·15-s − 0.242·17-s + 0.576·19-s − 1.70·21-s + 1.44·23-s − 0.695·25-s + 1.05·27-s − 1.33·29-s + 1.05·31-s + 0.500·33-s − 1.01·35-s + 0.574·37-s + 1.47·39-s − 0.195·41-s + 0.221·43-s + 0.0725·45-s + 0.354·47-s + 2.36·49-s + 0.226·51-s + 0.366·53-s + 0.296·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8024\)    =    \(2^{3} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8024} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8024,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;17,\;59\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;17,\;59\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
17 \( 1 + T \)
59 \( 1 + T \)
good3 \( 1 + 1.61T + 3T^{2} \)
5 \( 1 + 1.23T + 5T^{2} \)
7 \( 1 - 4.85T + 7T^{2} \)
11 \( 1 + 1.78T + 11T^{2} \)
13 \( 1 + 5.69T + 13T^{2} \)
19 \( 1 - 2.51T + 19T^{2} \)
23 \( 1 - 6.91T + 23T^{2} \)
29 \( 1 + 7.19T + 29T^{2} \)
31 \( 1 - 5.87T + 31T^{2} \)
37 \( 1 - 3.49T + 37T^{2} \)
41 \( 1 + 1.25T + 41T^{2} \)
43 \( 1 - 1.45T + 43T^{2} \)
47 \( 1 - 2.43T + 47T^{2} \)
53 \( 1 - 2.66T + 53T^{2} \)
61 \( 1 + 8.88T + 61T^{2} \)
67 \( 1 + 3.95T + 67T^{2} \)
71 \( 1 + 2.75T + 71T^{2} \)
73 \( 1 + 9.97T + 73T^{2} \)
79 \( 1 - 1.01T + 79T^{2} \)
83 \( 1 + 2.01T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 3.33T + 97T^{2} \)
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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

−7.60038914995642955828233394704, −6.97428462843155358734926137507, −5.85940844962796229312983790500, −5.26226668989182782100387067433, −4.80861716951887975877376126793, −4.29087142175975616833368605163, −2.98684290584353278522264295097, −2.18662360875008823934130395885, −1.09358943032154410070140605365, 0, 1.09358943032154410070140605365, 2.18662360875008823934130395885, 2.98684290584353278522264295097, 4.29087142175975616833368605163, 4.80861716951887975877376126793, 5.26226668989182782100387067433, 5.85940844962796229312983790500, 6.97428462843155358734926137507, 7.60038914995642955828233394704

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