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  + 1.25·3-s + 3.93·5-s − 3.56·7-s − 1.42·9-s − 3.07·11-s + 2.74·13-s + 4.94·15-s − 17-s + 2.16·19-s − 4.47·21-s − 5.55·23-s + 10.4·25-s − 5.55·27-s − 2.25·29-s − 0.00422·31-s − 3.86·33-s − 14.0·35-s + 5.75·37-s + 3.45·39-s − 2.70·41-s − 11.1·43-s − 5.59·45-s + 3.90·47-s + 5.70·49-s − 1.25·51-s − 6.71·53-s − 12.1·55-s + ⋯
L(s)  = 1  + 0.725·3-s + 1.75·5-s − 1.34·7-s − 0.473·9-s − 0.928·11-s + 0.762·13-s + 1.27·15-s − 0.242·17-s + 0.496·19-s − 0.977·21-s − 1.15·23-s + 2.09·25-s − 1.06·27-s − 0.419·29-s − 0.000758·31-s − 0.673·33-s − 2.37·35-s + 0.946·37-s + 0.553·39-s − 0.422·41-s − 1.69·43-s − 0.833·45-s + 0.569·47-s + 0.814·49-s − 0.175·51-s − 0.922·53-s − 1.63·55-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 \( 1 - 1.25T + 3T^{2} \)
5 \( 1 - 3.93T + 5T^{2} \)
7 \( 1 + 3.56T + 7T^{2} \)
11 \( 1 + 3.07T + 11T^{2} \)
13 \( 1 - 2.74T + 13T^{2} \)
19 \( 1 - 2.16T + 19T^{2} \)
23 \( 1 + 5.55T + 23T^{2} \)
29 \( 1 + 2.25T + 29T^{2} \)
31 \( 1 + 0.00422T + 31T^{2} \)
37 \( 1 - 5.75T + 37T^{2} \)
41 \( 1 + 2.70T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 - 3.90T + 47T^{2} \)
53 \( 1 + 6.71T + 53T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 + 5.44T + 67T^{2} \)
71 \( 1 + 3.00T + 71T^{2} \)
73 \( 1 + 0.207T + 73T^{2} \)
79 \( 1 + 3.57T + 79T^{2} \)
83 \( 1 + 4.16T + 83T^{2} \)
89 \( 1 - 5.88T + 89T^{2} \)
97 \( 1 + 9.10T + 97T^{2} \)
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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

−7.52788978205686659760686879770, −6.59398927872605937866585336157, −5.98358238128916350385301715642, −5.73831318444347144639312799020, −4.77582833614876706353809958604, −3.55353037819134240641169787807, −2.98562660362467806355966200537, −2.36560868754025837701214428061, −1.55335398052620210537217118422, 0, 1.55335398052620210537217118422, 2.36560868754025837701214428061, 2.98562660362467806355966200537, 3.55353037819134240641169787807, 4.77582833614876706353809958604, 5.73831318444347144639312799020, 5.98358238128916350385301715642, 6.59398927872605937866585336157, 7.52788978205686659760686879770

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