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  + 2.77·3-s − 1.91·5-s − 1.46·7-s + 4.69·9-s − 2.45·11-s + 0.468·13-s − 5.30·15-s − 17-s + 1.59·19-s − 4.05·21-s + 2.10·23-s − 1.34·25-s + 4.71·27-s − 1.76·29-s − 2.94·31-s − 6.80·33-s + 2.79·35-s + 2.38·37-s + 1.30·39-s + 6.96·41-s + 6.83·43-s − 8.98·45-s + 3.84·47-s − 4.86·49-s − 2.77·51-s − 3.98·53-s + 4.68·55-s + ⋯
L(s)  = 1  + 1.60·3-s − 0.855·5-s − 0.552·7-s + 1.56·9-s − 0.739·11-s + 0.129·13-s − 1.37·15-s − 0.242·17-s + 0.366·19-s − 0.885·21-s + 0.439·23-s − 0.268·25-s + 0.907·27-s − 0.327·29-s − 0.528·31-s − 1.18·33-s + 0.472·35-s + 0.392·37-s + 0.208·39-s + 1.08·41-s + 1.04·43-s − 1.33·45-s + 0.560·47-s − 0.694·49-s − 0.388·51-s − 0.547·53-s + 0.632·55-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 \( 1 - 2.77T + 3T^{2} \)
5 \( 1 + 1.91T + 5T^{2} \)
7 \( 1 + 1.46T + 7T^{2} \)
11 \( 1 + 2.45T + 11T^{2} \)
13 \( 1 - 0.468T + 13T^{2} \)
19 \( 1 - 1.59T + 19T^{2} \)
23 \( 1 - 2.10T + 23T^{2} \)
29 \( 1 + 1.76T + 29T^{2} \)
31 \( 1 + 2.94T + 31T^{2} \)
37 \( 1 - 2.38T + 37T^{2} \)
41 \( 1 - 6.96T + 41T^{2} \)
43 \( 1 - 6.83T + 43T^{2} \)
47 \( 1 - 3.84T + 47T^{2} \)
53 \( 1 + 3.98T + 53T^{2} \)
61 \( 1 + 2.87T + 61T^{2} \)
67 \( 1 + 9.15T + 67T^{2} \)
71 \( 1 + 16.4T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 6.65T + 79T^{2} \)
83 \( 1 + 7.60T + 83T^{2} \)
89 \( 1 - 3.95T + 89T^{2} \)
97 \( 1 + 13.2T + 97T^{2} \)
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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

−7.58798372753205585651040276126, −7.21671964857153986085712813737, −6.21061536284085372129318779386, −5.32470372443339334077290699526, −4.25425971568649581408463121134, −3.86165852959627289284261896262, −2.95634980792359799494599824881, −2.61254204478258227348005407589, −1.44540929860504133809559183375, 0, 1.44540929860504133809559183375, 2.61254204478258227348005407589, 2.95634980792359799494599824881, 3.86165852959627289284261896262, 4.25425971568649581408463121134, 5.32470372443339334077290699526, 6.21061536284085372129318779386, 7.21671964857153986085712813737, 7.58798372753205585651040276126

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