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 + 1.02·3-s + 4-s − 2.01·5-s − 1.02·6-s + 2.64·7-s − 8-s − 1.94·9-s + 2.01·10-s + 4.28·11-s + 1.02·12-s + 3.49·13-s − 2.64·14-s − 2.07·15-s + 16-s + 3.80·17-s + 1.94·18-s − 19-s − 2.01·20-s + 2.72·21-s − 4.28·22-s − 6.08·23-s − 1.02·24-s − 0.933·25-s − 3.49·26-s − 5.08·27-s + 2.64·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.594·3-s + 0.5·4-s − 0.901·5-s − 0.420·6-s + 0.999·7-s − 0.353·8-s − 0.647·9-s + 0.637·10-s + 1.29·11-s + 0.297·12-s + 0.969·13-s − 0.706·14-s − 0.535·15-s + 0.250·16-s + 0.922·17-s + 0.457·18-s − 0.229·19-s − 0.450·20-s + 0.593·21-s − 0.913·22-s − 1.26·23-s − 0.210·24-s − 0.186·25-s − 0.685·26-s − 0.978·27-s + 0.499·28-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 \( 1 - 1.02T + 3T^{2} \)
5 \( 1 + 2.01T + 5T^{2} \)
7 \( 1 - 2.64T + 7T^{2} \)
11 \( 1 - 4.28T + 11T^{2} \)
13 \( 1 - 3.49T + 13T^{2} \)
17 \( 1 - 3.80T + 17T^{2} \)
23 \( 1 + 6.08T + 23T^{2} \)
29 \( 1 + 3.51T + 29T^{2} \)
31 \( 1 + 8.03T + 31T^{2} \)
37 \( 1 + 4.42T + 37T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 - 0.137T + 43T^{2} \)
47 \( 1 - 5.03T + 47T^{2} \)
53 \( 1 + 0.646T + 53T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 - 3.10T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 9.96T + 71T^{2} \)
73 \( 1 + 9.76T + 73T^{2} \)
79 \( 1 + 9.82T + 79T^{2} \)
83 \( 1 - 4.59T + 83T^{2} \)
89 \( 1 - 3.64T + 89T^{2} \)
97 \( 1 - 1.87T + 97T^{2} \)
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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

−7.73888646185286952093489432843, −7.16440685182052882796255795463, −6.03791073071453345005071242270, −5.68124253652827794378301627917, −4.38296803934087060522564119926, −3.76332078155394826079445156566, −3.22460937058552818112296497557, −1.94043128987454882414675650016, −1.37549920804921051002502559508, 0, 1.37549920804921051002502559508, 1.94043128987454882414675650016, 3.22460937058552818112296497557, 3.76332078155394826079445156566, 4.38296803934087060522564119926, 5.68124253652827794378301627917, 6.03791073071453345005071242270, 7.16440685182052882796255795463, 7.73888646185286952093489432843

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