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  − 1.49·3-s − 3.21·5-s − 1.95·7-s − 0.755·9-s + 5.72·11-s − 6.02·13-s + 4.81·15-s − 17-s + 2.70·19-s + 2.92·21-s − 1.14·23-s + 5.32·25-s + 5.62·27-s − 2.95·29-s − 3.60·31-s − 8.57·33-s + 6.26·35-s + 1.95·37-s + 9.02·39-s − 5.74·41-s + 7.26·43-s + 2.42·45-s − 8.00·47-s − 3.19·49-s + 1.49·51-s + 13.8·53-s − 18.3·55-s + ⋯
L(s)  = 1  − 0.864·3-s − 1.43·5-s − 0.737·7-s − 0.251·9-s + 1.72·11-s − 1.67·13-s + 1.24·15-s − 0.242·17-s + 0.620·19-s + 0.637·21-s − 0.239·23-s + 1.06·25-s + 1.08·27-s − 0.548·29-s − 0.647·31-s − 1.49·33-s + 1.05·35-s + 0.321·37-s + 1.44·39-s − 0.897·41-s + 1.10·43-s + 0.361·45-s − 1.16·47-s − 0.456·49-s + 0.209·51-s + 1.90·53-s − 2.48·55-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 \( 1 + 1.49T + 3T^{2} \)
5 \( 1 + 3.21T + 5T^{2} \)
7 \( 1 + 1.95T + 7T^{2} \)
11 \( 1 - 5.72T + 11T^{2} \)
13 \( 1 + 6.02T + 13T^{2} \)
19 \( 1 - 2.70T + 19T^{2} \)
23 \( 1 + 1.14T + 23T^{2} \)
29 \( 1 + 2.95T + 29T^{2} \)
31 \( 1 + 3.60T + 31T^{2} \)
37 \( 1 - 1.95T + 37T^{2} \)
41 \( 1 + 5.74T + 41T^{2} \)
43 \( 1 - 7.26T + 43T^{2} \)
47 \( 1 + 8.00T + 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
61 \( 1 + 12.5T + 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 - 9.36T + 73T^{2} \)
79 \( 1 - 16.8T + 79T^{2} \)
83 \( 1 + 0.256T + 83T^{2} \)
89 \( 1 - 5.77T + 89T^{2} \)
97 \( 1 - 4.67T + 97T^{2} \)
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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

−7.38943800178146822722263429343, −6.74168179886972266159562225568, −6.30524466361464006858253703838, −5.28651333694159655605838188474, −4.72135342929751951560889946476, −3.81766176815685981396665812129, −3.41319824784280368781746140693, −2.26836698524869141937595489246, −0.818790509958938628049838663111, 0, 0.818790509958938628049838663111, 2.26836698524869141937595489246, 3.41319824784280368781746140693, 3.81766176815685981396665812129, 4.72135342929751951560889946476, 5.28651333694159655605838188474, 6.30524466361464006858253703838, 6.74168179886972266159562225568, 7.38943800178146822722263429343

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