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  + 3.05·3-s − 3.42·5-s + 2.67·7-s + 6.34·9-s + 2.12·11-s − 1.43·13-s − 10.4·15-s − 17-s − 4.85·19-s + 8.16·21-s − 9.15·23-s + 6.74·25-s + 10.2·27-s − 4.91·29-s − 5.96·31-s + 6.47·33-s − 9.15·35-s − 3.49·37-s − 4.37·39-s + 4.45·41-s − 9.79·43-s − 21.7·45-s − 12.3·47-s + 0.140·49-s − 3.05·51-s − 4.08·53-s − 7.26·55-s + ⋯
L(s)  = 1  + 1.76·3-s − 1.53·5-s + 1.01·7-s + 2.11·9-s + 0.639·11-s − 0.397·13-s − 2.70·15-s − 0.242·17-s − 1.11·19-s + 1.78·21-s − 1.90·23-s + 1.34·25-s + 1.96·27-s − 0.912·29-s − 1.07·31-s + 1.12·33-s − 1.54·35-s − 0.573·37-s − 0.700·39-s + 0.696·41-s − 1.49·43-s − 3.23·45-s − 1.80·47-s + 0.0201·49-s − 0.427·51-s − 0.561·53-s − 0.979·55-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 \( 1 - 3.05T + 3T^{2} \)
5 \( 1 + 3.42T + 5T^{2} \)
7 \( 1 - 2.67T + 7T^{2} \)
11 \( 1 - 2.12T + 11T^{2} \)
13 \( 1 + 1.43T + 13T^{2} \)
19 \( 1 + 4.85T + 19T^{2} \)
23 \( 1 + 9.15T + 23T^{2} \)
29 \( 1 + 4.91T + 29T^{2} \)
31 \( 1 + 5.96T + 31T^{2} \)
37 \( 1 + 3.49T + 37T^{2} \)
41 \( 1 - 4.45T + 41T^{2} \)
43 \( 1 + 9.79T + 43T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 + 4.08T + 53T^{2} \)
61 \( 1 - 8.60T + 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 7.95T + 73T^{2} \)
79 \( 1 + 8.63T + 79T^{2} \)
83 \( 1 - 4.37T + 83T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−7.77253204486810192417182273006, −7.16382142455720356294827250152, −6.39434092774693352529581987328, −5.02607873722602437211249712086, −4.30601851315833275655743262898, −3.83102478758924744253953450562, −3.33568468412394009955645279069, −2.12319218628916696859647709987, −1.69834128185127170096220065441, 0, 1.69834128185127170096220065441, 2.12319218628916696859647709987, 3.33568468412394009955645279069, 3.83102478758924744253953450562, 4.30601851315833275655743262898, 5.02607873722602437211249712086, 6.39434092774693352529581987328, 7.16382142455720356294827250152, 7.77253204486810192417182273006

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