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  + 1.45·3-s − 0.727·5-s + 2.11·7-s − 0.880·9-s − 2.82·11-s + 3.00·13-s − 1.05·15-s − 17-s − 1.63·19-s + 3.08·21-s + 6.69·23-s − 4.47·25-s − 5.64·27-s − 3.98·29-s − 6.44·31-s − 4.11·33-s − 1.53·35-s + 1.41·37-s + 4.37·39-s − 4.60·41-s + 4.91·43-s + 0.640·45-s − 1.35·47-s − 2.52·49-s − 1.45·51-s + 8.50·53-s + 2.05·55-s + ⋯
L(s)  = 1  + 0.840·3-s − 0.325·5-s + 0.799·7-s − 0.293·9-s − 0.851·11-s + 0.832·13-s − 0.273·15-s − 0.242·17-s − 0.375·19-s + 0.672·21-s + 1.39·23-s − 0.894·25-s − 1.08·27-s − 0.739·29-s − 1.15·31-s − 0.715·33-s − 0.260·35-s + 0.232·37-s + 0.699·39-s − 0.718·41-s + 0.749·43-s + 0.0954·45-s − 0.197·47-s − 0.360·49-s − 0.203·51-s + 1.16·53-s + 0.277·55-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 \( 1 - 1.45T + 3T^{2} \)
5 \( 1 + 0.727T + 5T^{2} \)
7 \( 1 - 2.11T + 7T^{2} \)
11 \( 1 + 2.82T + 11T^{2} \)
13 \( 1 - 3.00T + 13T^{2} \)
19 \( 1 + 1.63T + 19T^{2} \)
23 \( 1 - 6.69T + 23T^{2} \)
29 \( 1 + 3.98T + 29T^{2} \)
31 \( 1 + 6.44T + 31T^{2} \)
37 \( 1 - 1.41T + 37T^{2} \)
41 \( 1 + 4.60T + 41T^{2} \)
43 \( 1 - 4.91T + 43T^{2} \)
47 \( 1 + 1.35T + 47T^{2} \)
53 \( 1 - 8.50T + 53T^{2} \)
61 \( 1 - 0.0785T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 - 3.96T + 71T^{2} \)
73 \( 1 + 9.33T + 73T^{2} \)
79 \( 1 + 5.60T + 79T^{2} \)
83 \( 1 + 9.36T + 83T^{2} \)
89 \( 1 + 14.7T + 89T^{2} \)
97 \( 1 - 1.70T + 97T^{2} \)
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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

−7.58704878892241521225590112139, −7.10558685804273816962441724516, −5.94169370387910173744867607762, −5.44302522683786817785685759192, −4.57298096712127996413027202477, −3.79817846045432634690611282667, −3.10033686201591463016165063503, −2.28831263630185087471552012819, −1.46519553442032322211272110548, 0, 1.46519553442032322211272110548, 2.28831263630185087471552012819, 3.10033686201591463016165063503, 3.79817846045432634690611282667, 4.57298096712127996413027202477, 5.44302522683786817785685759192, 5.94169370387910173744867607762, 7.10558685804273816962441724516, 7.58704878892241521225590112139

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