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 − 2.74·3-s + 4-s + 0.116·5-s − 2.74·6-s − 0.997·7-s + 8-s + 4.51·9-s + 0.116·10-s + 3.52·11-s − 2.74·12-s + 1.83·13-s − 0.997·14-s − 0.320·15-s + 16-s − 2.74·17-s + 4.51·18-s + 19-s + 0.116·20-s + 2.73·21-s + 3.52·22-s + 3.51·23-s − 2.74·24-s − 4.98·25-s + 1.83·26-s − 4.14·27-s − 0.997·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.58·3-s + 0.5·4-s + 0.0523·5-s − 1.11·6-s − 0.376·7-s + 0.353·8-s + 1.50·9-s + 0.0369·10-s + 1.06·11-s − 0.791·12-s + 0.509·13-s − 0.266·14-s − 0.0827·15-s + 0.250·16-s − 0.666·17-s + 1.06·18-s + 0.229·19-s + 0.0261·20-s + 0.596·21-s + 0.751·22-s + 0.732·23-s − 0.559·24-s − 0.997·25-s + 0.360·26-s − 0.797·27-s − 0.188·28-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 \( 1 + 2.74T + 3T^{2} \)
5 \( 1 - 0.116T + 5T^{2} \)
7 \( 1 + 0.997T + 7T^{2} \)
11 \( 1 - 3.52T + 11T^{2} \)
13 \( 1 - 1.83T + 13T^{2} \)
17 \( 1 + 2.74T + 17T^{2} \)
23 \( 1 - 3.51T + 23T^{2} \)
29 \( 1 + 2.84T + 29T^{2} \)
31 \( 1 + 7.76T + 31T^{2} \)
37 \( 1 - 1.28T + 37T^{2} \)
41 \( 1 + 5.75T + 41T^{2} \)
43 \( 1 + 9.54T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + 1.96T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 - 8.40T + 67T^{2} \)
71 \( 1 - 5.48T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 - 2.10T + 79T^{2} \)
83 \( 1 - 0.895T + 83T^{2} \)
89 \( 1 - 4.04T + 89T^{2} \)
97 \( 1 - 2.54T + 97T^{2} \)
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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

−6.94832058691111062537792795853, −6.61360227817784836126547784912, −6.17804955613567866009096635333, −5.20154674099959508198508555567, −5.05642831073251120249652456434, −3.83336435033402837276038832027, −3.58322125350961565825471623410, −2.08094773754668386852956904894, −1.22160400724322805009456388189, 0, 1.22160400724322805009456388189, 2.08094773754668386852956904894, 3.58322125350961565825471623410, 3.83336435033402837276038832027, 5.05642831073251120249652456434, 5.20154674099959508198508555567, 6.17804955613567866009096635333, 6.61360227817784836126547784912, 6.94832058691111062537792795853

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