Properties

Degree 2
Conductor $ 19 \cdot 317 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.90·2-s + 0.192·3-s + 1.63·4-s + 0.483·5-s − 0.366·6-s + 3.15·7-s + 0.704·8-s − 2.96·9-s − 0.921·10-s + 0.275·11-s + 0.313·12-s − 1.15·13-s − 6.01·14-s + 0.0930·15-s − 4.60·16-s − 4.88·17-s + 5.64·18-s − 19-s + 0.788·20-s + 0.607·21-s − 0.525·22-s + 7.55·23-s + 0.135·24-s − 4.76·25-s + 2.20·26-s − 1.14·27-s + 5.14·28-s + ⋯
L(s)  = 1  − 1.34·2-s + 0.111·3-s + 0.815·4-s + 0.216·5-s − 0.149·6-s + 1.19·7-s + 0.249·8-s − 0.987·9-s − 0.291·10-s + 0.0832·11-s + 0.0905·12-s − 0.321·13-s − 1.60·14-s + 0.0240·15-s − 1.15·16-s − 1.18·17-s + 1.33·18-s − 0.229·19-s + 0.176·20-s + 0.132·21-s − 0.112·22-s + 1.57·23-s + 0.0276·24-s − 0.953·25-s + 0.433·26-s − 0.220·27-s + 0.971·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6023\)    =    \(19 \cdot 317\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6023} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6023,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{19,\;317\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{19,\;317\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad19 \( 1 + T \)
317 \( 1 + T \)
good2 \( 1 + 1.90T + 2T^{2} \)
3 \( 1 - 0.192T + 3T^{2} \)
5 \( 1 - 0.483T + 5T^{2} \)
7 \( 1 - 3.15T + 7T^{2} \)
11 \( 1 - 0.275T + 11T^{2} \)
13 \( 1 + 1.15T + 13T^{2} \)
17 \( 1 + 4.88T + 17T^{2} \)
23 \( 1 - 7.55T + 23T^{2} \)
29 \( 1 + 1.37T + 29T^{2} \)
31 \( 1 - 1.85T + 31T^{2} \)
37 \( 1 - 8.37T + 37T^{2} \)
41 \( 1 - 2.29T + 41T^{2} \)
43 \( 1 + 7.33T + 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 + 4.39T + 53T^{2} \)
59 \( 1 - 2.49T + 59T^{2} \)
61 \( 1 - 8.52T + 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + 6.40T + 71T^{2} \)
73 \( 1 - 8.31T + 73T^{2} \)
79 \( 1 + 1.19T + 79T^{2} \)
83 \( 1 + 4.01T + 83T^{2} \)
89 \( 1 + 14.4T + 89T^{2} \)
97 \( 1 - 3.23T + 97T^{2} \)
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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

−8.101276313480150221330288150412, −7.24547697866697237572654197439, −6.58395584391994109523580461916, −5.61543306920947170934614706765, −4.84007268043344802582511495706, −4.18326249147111962540206587396, −2.78649849728044321013664993230, −2.09698110094489776488385919025, −1.20282520903872912930234141402, 0, 1.20282520903872912930234141402, 2.09698110094489776488385919025, 2.78649849728044321013664993230, 4.18326249147111962540206587396, 4.84007268043344802582511495706, 5.61543306920947170934614706765, 6.58395584391994109523580461916, 7.24547697866697237572654197439, 8.101276313480150221330288150412

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