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  − 1.40·3-s + 3.37·5-s − 3.92·7-s − 1.02·9-s + 1.19·11-s + 1.76·13-s − 4.73·15-s − 17-s − 5.97·19-s + 5.51·21-s − 1.06·23-s + 6.38·25-s + 5.65·27-s + 5.79·29-s + 0.924·31-s − 1.67·33-s − 13.2·35-s − 6.65·37-s − 2.48·39-s + 8.89·41-s + 8.28·43-s − 3.46·45-s − 1.68·47-s + 8.40·49-s + 1.40·51-s + 3.41·53-s + 4.03·55-s + ⋯
L(s)  = 1  − 0.810·3-s + 1.50·5-s − 1.48·7-s − 0.342·9-s + 0.360·11-s + 0.490·13-s − 1.22·15-s − 0.242·17-s − 1.36·19-s + 1.20·21-s − 0.221·23-s + 1.27·25-s + 1.08·27-s + 1.07·29-s + 0.165·31-s − 0.292·33-s − 2.23·35-s − 1.09·37-s − 0.397·39-s + 1.38·41-s + 1.26·43-s − 0.517·45-s − 0.245·47-s + 1.20·49-s + 0.196·51-s + 0.469·53-s + 0.543·55-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 \( 1 + 1.40T + 3T^{2} \)
5 \( 1 - 3.37T + 5T^{2} \)
7 \( 1 + 3.92T + 7T^{2} \)
11 \( 1 - 1.19T + 11T^{2} \)
13 \( 1 - 1.76T + 13T^{2} \)
19 \( 1 + 5.97T + 19T^{2} \)
23 \( 1 + 1.06T + 23T^{2} \)
29 \( 1 - 5.79T + 29T^{2} \)
31 \( 1 - 0.924T + 31T^{2} \)
37 \( 1 + 6.65T + 37T^{2} \)
41 \( 1 - 8.89T + 41T^{2} \)
43 \( 1 - 8.28T + 43T^{2} \)
47 \( 1 + 1.68T + 47T^{2} \)
53 \( 1 - 3.41T + 53T^{2} \)
61 \( 1 + 6.47T + 61T^{2} \)
67 \( 1 + 1.63T + 67T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 12.4T + 73T^{2} \)
79 \( 1 + 11.1T + 79T^{2} \)
83 \( 1 - 9.00T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−7.09131431582211668028095539483, −6.50135987871663337937960707342, −6.02607211859839925800754328272, −5.82424505708533525695523842691, −4.80959928495458629465273737975, −3.95270195525383041524598344729, −2.91463566852338611146686557643, −2.31689394536841048899011693760, −1.15808945461239984875275686049, 0, 1.15808945461239984875275686049, 2.31689394536841048899011693760, 2.91463566852338611146686557643, 3.95270195525383041524598344729, 4.80959928495458629465273737975, 5.82424505708533525695523842691, 6.02607211859839925800754328272, 6.50135987871663337937960707342, 7.09131431582211668028095539483

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