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  + 1.04·3-s − 0.295·5-s + 1.92·7-s − 1.91·9-s + 3.27·11-s + 3.24·13-s − 0.307·15-s − 17-s − 1.86·19-s + 2.00·21-s − 2.35·23-s − 4.91·25-s − 5.12·27-s − 1.73·29-s − 0.135·31-s + 3.41·33-s − 0.567·35-s − 10.7·37-s + 3.38·39-s − 11.0·41-s − 6.38·43-s + 0.565·45-s − 6.90·47-s − 3.31·49-s − 1.04·51-s − 1.91·53-s − 0.968·55-s + ⋯
L(s)  = 1  + 0.601·3-s − 0.132·5-s + 0.725·7-s − 0.637·9-s + 0.988·11-s + 0.899·13-s − 0.0795·15-s − 0.242·17-s − 0.427·19-s + 0.436·21-s − 0.490·23-s − 0.982·25-s − 0.985·27-s − 0.321·29-s − 0.0242·31-s + 0.595·33-s − 0.0959·35-s − 1.76·37-s + 0.541·39-s − 1.73·41-s − 0.973·43-s + 0.0842·45-s − 1.00·47-s − 0.473·49-s − 0.145·51-s − 0.262·53-s − 0.130·55-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 \( 1 - 1.04T + 3T^{2} \)
5 \( 1 + 0.295T + 5T^{2} \)
7 \( 1 - 1.92T + 7T^{2} \)
11 \( 1 - 3.27T + 11T^{2} \)
13 \( 1 - 3.24T + 13T^{2} \)
19 \( 1 + 1.86T + 19T^{2} \)
23 \( 1 + 2.35T + 23T^{2} \)
29 \( 1 + 1.73T + 29T^{2} \)
31 \( 1 + 0.135T + 31T^{2} \)
37 \( 1 + 10.7T + 37T^{2} \)
41 \( 1 + 11.0T + 41T^{2} \)
43 \( 1 + 6.38T + 43T^{2} \)
47 \( 1 + 6.90T + 47T^{2} \)
53 \( 1 + 1.91T + 53T^{2} \)
61 \( 1 + 2.57T + 61T^{2} \)
67 \( 1 - 1.87T + 67T^{2} \)
71 \( 1 + 0.397T + 71T^{2} \)
73 \( 1 + 2.58T + 73T^{2} \)
79 \( 1 - 6.41T + 79T^{2} \)
83 \( 1 - 8.47T + 83T^{2} \)
89 \( 1 - 1.33T + 89T^{2} \)
97 \( 1 + 4.73T + 97T^{2} \)
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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

−7.67403585378128763668060503972, −6.73501190334554509604098384387, −6.20853403611180519429270418326, −5.37288388446002896807535377979, −4.61294712019994773978348594562, −3.63818145030939087296013140765, −3.38631906670358356715691175552, −2.01524524246251579197694887567, −1.58041565350037373744932295601, 0, 1.58041565350037373744932295601, 2.01524524246251579197694887567, 3.38631906670358356715691175552, 3.63818145030939087296013140765, 4.61294712019994773978348594562, 5.37288388446002896807535377979, 6.20853403611180519429270418326, 6.73501190334554509604098384387, 7.67403585378128763668060503972

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