Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.61·3-s + 4-s − 2.67·5-s − 1.61·6-s − 0.794·7-s − 8-s − 0.391·9-s + 2.67·10-s − 2.37·11-s + 1.61·12-s − 0.106·13-s + 0.794·14-s − 4.32·15-s + 16-s + 2.83·17-s + 0.391·18-s − 19-s − 2.67·20-s − 1.28·21-s + 2.37·22-s + 6.31·23-s − 1.61·24-s + 2.16·25-s + 0.106·26-s − 5.47·27-s − 0.794·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.932·3-s + 0.5·4-s − 1.19·5-s − 0.659·6-s − 0.300·7-s − 0.353·8-s − 0.130·9-s + 0.846·10-s − 0.714·11-s + 0.466·12-s − 0.0294·13-s + 0.212·14-s − 1.11·15-s + 0.250·16-s + 0.688·17-s + 0.0923·18-s − 0.229·19-s − 0.598·20-s − 0.279·21-s + 0.505·22-s + 1.31·23-s − 0.329·24-s + 0.432·25-s + 0.0208·26-s − 1.05·27-s − 0.150·28-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 \( 1 - 1.61T + 3T^{2} \)
5 \( 1 + 2.67T + 5T^{2} \)
7 \( 1 + 0.794T + 7T^{2} \)
11 \( 1 + 2.37T + 11T^{2} \)
13 \( 1 + 0.106T + 13T^{2} \)
17 \( 1 - 2.83T + 17T^{2} \)
23 \( 1 - 6.31T + 23T^{2} \)
29 \( 1 - 1.48T + 29T^{2} \)
31 \( 1 - 7.04T + 31T^{2} \)
37 \( 1 - 8.75T + 37T^{2} \)
41 \( 1 + 9.87T + 41T^{2} \)
43 \( 1 - 0.390T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 3.80T + 53T^{2} \)
59 \( 1 - 7.62T + 59T^{2} \)
61 \( 1 + 7.41T + 61T^{2} \)
67 \( 1 + 7.63T + 67T^{2} \)
71 \( 1 - 5.42T + 71T^{2} \)
73 \( 1 + 4.28T + 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + 3.26T + 89T^{2} \)
97 \( 1 + 9.02T + 97T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.81804642227327553581212452024, −7.10537044504153298168582023309, −6.34728502982198250648838528294, −5.39651871307871485475005946019, −4.52431852454437651233180345532, −3.61454305458808009052602252078, −2.99965415142208800589618264216, −2.45478797058236335170889221052, −1.09636677486236666189832742111, 0, 1.09636677486236666189832742111, 2.45478797058236335170889221052, 2.99965415142208800589618264216, 3.61454305458808009052602252078, 4.52431852454437651233180345532, 5.39651871307871485475005946019, 6.34728502982198250648838528294, 7.10537044504153298168582023309, 7.81804642227327553581212452024

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