Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
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-s + 0.920·3-s + 4-s − 2.78·5-s + 0.920·6-s − 0.210·7-s + 8-s − 2.15·9-s − 2.78·10-s + 5.28·11-s + 0.920·12-s + 1.36·13-s − 0.210·14-s − 2.56·15-s + 16-s − 3.62·17-s − 2.15·18-s + 19-s − 2.78·20-s − 0.193·21-s + 5.28·22-s − 4.33·23-s + 0.920·24-s + 2.74·25-s + 1.36·26-s − 4.74·27-s − 0.210·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.531·3-s + 0.5·4-s − 1.24·5-s + 0.375·6-s − 0.0794·7-s + 0.353·8-s − 0.717·9-s − 0.880·10-s + 1.59·11-s + 0.265·12-s + 0.379·13-s − 0.0561·14-s − 0.661·15-s + 0.250·16-s − 0.879·17-s − 0.507·18-s + 0.229·19-s − 0.622·20-s − 0.0422·21-s + 1.12·22-s − 0.903·23-s + 0.187·24-s + 0.548·25-s + 0.268·26-s − 0.912·27-s − 0.0397·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8018\)    =    \(2 \cdot 19 \cdot 211\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8018} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;19,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
19 \( 1 - T \)
211 \( 1 + T \)
good3 \( 1 - 0.920T + 3T^{2} \)
5 \( 1 + 2.78T + 5T^{2} \)
7 \( 1 + 0.210T + 7T^{2} \)
11 \( 1 - 5.28T + 11T^{2} \)
13 \( 1 - 1.36T + 13T^{2} \)
17 \( 1 + 3.62T + 17T^{2} \)
23 \( 1 + 4.33T + 23T^{2} \)
29 \( 1 + 9.10T + 29T^{2} \)
31 \( 1 - 5.99T + 31T^{2} \)
37 \( 1 - 0.893T + 37T^{2} \)
41 \( 1 - 2.52T + 41T^{2} \)
43 \( 1 - 3.37T + 43T^{2} \)
47 \( 1 + 7.73T + 47T^{2} \)
53 \( 1 - 7.53T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 8.16T + 61T^{2} \)
67 \( 1 + 1.47T + 67T^{2} \)
71 \( 1 - 7.74T + 71T^{2} \)
73 \( 1 + 9.00T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 7.51T + 83T^{2} \)
89 \( 1 + 8.00T + 89T^{2} \)
97 \( 1 + 7.56T + 97T^{2} \)
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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

−7.50896919620148446988305458179, −6.73289442227189967352896947859, −6.16864819531026324878173236282, −5.38243337940429976360333785462, −4.20314397015843654610155318411, −4.02148946841000375954018736741, −3.34541575484793216309033994095, −2.46822679184992960323502481886, −1.42912102220169143540537570011, 0, 1.42912102220169143540537570011, 2.46822679184992960323502481886, 3.34541575484793216309033994095, 4.02148946841000375954018736741, 4.20314397015843654610155318411, 5.38243337940429976360333785462, 6.16864819531026324878173236282, 6.73289442227189967352896947859, 7.50896919620148446988305458179

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