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 − 3.31·3-s + 4-s − 0.532·5-s + 3.31·6-s − 2.51·7-s − 8-s + 7.99·9-s + 0.532·10-s + 6.00·11-s − 3.31·12-s + 4.31·13-s + 2.51·14-s + 1.76·15-s + 16-s + 2.00·17-s − 7.99·18-s − 19-s − 0.532·20-s + 8.32·21-s − 6.00·22-s − 2.59·23-s + 3.31·24-s − 4.71·25-s − 4.31·26-s − 16.5·27-s − 2.51·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.91·3-s + 0.5·4-s − 0.238·5-s + 1.35·6-s − 0.949·7-s − 0.353·8-s + 2.66·9-s + 0.168·10-s + 1.80·11-s − 0.957·12-s + 1.19·13-s + 0.671·14-s + 0.455·15-s + 0.250·16-s + 0.485·17-s − 1.88·18-s − 0.229·19-s − 0.119·20-s + 1.81·21-s − 1.27·22-s − 0.540·23-s + 0.676·24-s − 0.943·25-s − 0.846·26-s − 3.18·27-s − 0.474·28-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 \( 1 + 3.31T + 3T^{2} \)
5 \( 1 + 0.532T + 5T^{2} \)
7 \( 1 + 2.51T + 7T^{2} \)
11 \( 1 - 6.00T + 11T^{2} \)
13 \( 1 - 4.31T + 13T^{2} \)
17 \( 1 - 2.00T + 17T^{2} \)
23 \( 1 + 2.59T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 - 5.47T + 31T^{2} \)
37 \( 1 + 9.44T + 37T^{2} \)
41 \( 1 - 0.879T + 41T^{2} \)
43 \( 1 - 2.40T + 43T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 14.3T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 + 9.80T + 67T^{2} \)
71 \( 1 + 2.81T + 71T^{2} \)
73 \( 1 - 3.87T + 73T^{2} \)
79 \( 1 + 1.12T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 - 7.84T + 97T^{2} \)
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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

−7.20336359622919751035887992058, −6.49818491143245938561302279651, −6.32845277542448071290999033297, −5.81130724699252900984609955776, −4.68385730577017968817921389708, −3.99727568492586585691199179429, −3.29045717730358073923166023330, −1.59023857009934958213518791542, −1.03631530460901430805135592834, 0, 1.03631530460901430805135592834, 1.59023857009934958213518791542, 3.29045717730358073923166023330, 3.99727568492586585691199179429, 4.68385730577017968817921389708, 5.81130724699252900984609955776, 6.32845277542448071290999033297, 6.49818491143245938561302279651, 7.20336359622919751035887992058

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