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 − 0.894·3-s + 4-s − 0.0669·5-s − 0.894·6-s + 1.42·7-s + 8-s − 2.19·9-s − 0.0669·10-s − 3.23·11-s − 0.894·12-s + 4.17·13-s + 1.42·14-s + 0.0599·15-s + 16-s − 0.463·17-s − 2.19·18-s + 19-s − 0.0669·20-s − 1.27·21-s − 3.23·22-s − 2.96·23-s − 0.894·24-s − 4.99·25-s + 4.17·26-s + 4.65·27-s + 1.42·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.516·3-s + 0.5·4-s − 0.0299·5-s − 0.365·6-s + 0.540·7-s + 0.353·8-s − 0.733·9-s − 0.0211·10-s − 0.974·11-s − 0.258·12-s + 1.15·13-s + 0.382·14-s + 0.0154·15-s + 0.250·16-s − 0.112·17-s − 0.518·18-s + 0.229·19-s − 0.0149·20-s − 0.279·21-s − 0.689·22-s − 0.617·23-s − 0.182·24-s − 0.999·25-s + 0.819·26-s + 0.895·27-s + 0.270·28-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 \( 1 + 0.894T + 3T^{2} \)
5 \( 1 + 0.0669T + 5T^{2} \)
7 \( 1 - 1.42T + 7T^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
13 \( 1 - 4.17T + 13T^{2} \)
17 \( 1 + 0.463T + 17T^{2} \)
23 \( 1 + 2.96T + 23T^{2} \)
29 \( 1 - 5.89T + 29T^{2} \)
31 \( 1 + 5.02T + 31T^{2} \)
37 \( 1 + 5.53T + 37T^{2} \)
41 \( 1 - 3.26T + 41T^{2} \)
43 \( 1 - 0.640T + 43T^{2} \)
47 \( 1 - 0.940T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 - 3.02T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 + 7.46T + 67T^{2} \)
71 \( 1 + 2.88T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 1.25T + 83T^{2} \)
89 \( 1 + 3.46T + 89T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−7.45972333187088991726008034914, −6.55122532375464468048464325790, −5.93682704262252002199688813798, −5.42940609949994448286765565631, −4.81369658380621443657557288955, −3.95799066707751006092068112338, −3.19242887913744645865452132067, −2.34256483258567364599128122881, −1.36859176921252077902887965310, 0, 1.36859176921252077902887965310, 2.34256483258567364599128122881, 3.19242887913744645865452132067, 3.95799066707751006092068112338, 4.81369658380621443657557288955, 5.42940609949994448286765565631, 5.93682704262252002199688813798, 6.55122532375464468048464325790, 7.45972333187088991726008034914

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