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 − 0.350·3-s + 4-s − 0.178·5-s + 0.350·6-s − 3.90·7-s − 8-s − 2.87·9-s + 0.178·10-s − 1.52·11-s − 0.350·12-s − 0.315·13-s + 3.90·14-s + 0.0624·15-s + 16-s + 6.70·17-s + 2.87·18-s − 19-s − 0.178·20-s + 1.36·21-s + 1.52·22-s − 3.64·23-s + 0.350·24-s − 4.96·25-s + 0.315·26-s + 2.05·27-s − 3.90·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.202·3-s + 0.5·4-s − 0.0797·5-s + 0.143·6-s − 1.47·7-s − 0.353·8-s − 0.959·9-s + 0.0563·10-s − 0.460·11-s − 0.101·12-s − 0.0874·13-s + 1.04·14-s + 0.0161·15-s + 0.250·16-s + 1.62·17-s + 0.678·18-s − 0.229·19-s − 0.0398·20-s + 0.298·21-s + 0.325·22-s − 0.759·23-s + 0.0715·24-s − 0.993·25-s + 0.0618·26-s + 0.396·27-s − 0.738·28-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 \( 1 + 0.350T + 3T^{2} \)
5 \( 1 + 0.178T + 5T^{2} \)
7 \( 1 + 3.90T + 7T^{2} \)
11 \( 1 + 1.52T + 11T^{2} \)
13 \( 1 + 0.315T + 13T^{2} \)
17 \( 1 - 6.70T + 17T^{2} \)
23 \( 1 + 3.64T + 23T^{2} \)
29 \( 1 - 7.13T + 29T^{2} \)
31 \( 1 - 7.92T + 31T^{2} \)
37 \( 1 - 7.58T + 37T^{2} \)
41 \( 1 + 5.82T + 41T^{2} \)
43 \( 1 + 4.70T + 43T^{2} \)
47 \( 1 + 1.67T + 47T^{2} \)
53 \( 1 - 3.11T + 53T^{2} \)
59 \( 1 + 13.1T + 59T^{2} \)
61 \( 1 - 3.19T + 61T^{2} \)
67 \( 1 - 2.05T + 67T^{2} \)
71 \( 1 - 5.71T + 71T^{2} \)
73 \( 1 - 5.78T + 73T^{2} \)
79 \( 1 + 5.62T + 79T^{2} \)
83 \( 1 - 5.96T + 83T^{2} \)
89 \( 1 - 6.61T + 89T^{2} \)
97 \( 1 - 5.63T + 97T^{2} \)
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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

−7.76533682008439401860134800469, −6.65146247278470681950961566053, −6.22118075500722300222193628552, −5.69470812841626748604717502110, −4.76996564577323324025714536833, −3.56801113907900126264122921585, −3.05238024638914152179724343676, −2.33075897667473427730153638521, −0.912140549070592773162440665721, 0, 0.912140549070592773162440665721, 2.33075897667473427730153638521, 3.05238024638914152179724343676, 3.56801113907900126264122921585, 4.76996564577323324025714536833, 5.69470812841626748604717502110, 6.22118075500722300222193628552, 6.65146247278470681950961566053, 7.76533682008439401860134800469

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