Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3.39·3-s + 4-s + 3.61·5-s + 3.39·6-s − 0.821·7-s + 8-s + 8.51·9-s + 3.61·10-s − 2.67·11-s + 3.39·12-s + 3.43·13-s − 0.821·14-s + 12.2·15-s + 16-s − 5.28·17-s + 8.51·18-s − 19-s + 3.61·20-s − 2.78·21-s − 2.67·22-s − 6.30·23-s + 3.39·24-s + 8.09·25-s + 3.43·26-s + 18.7·27-s − 0.821·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.95·3-s + 0.5·4-s + 1.61·5-s + 1.38·6-s − 0.310·7-s + 0.353·8-s + 2.83·9-s + 1.14·10-s − 0.807·11-s + 0.979·12-s + 0.953·13-s − 0.219·14-s + 3.17·15-s + 0.250·16-s − 1.28·17-s + 2.00·18-s − 0.229·19-s + 0.809·20-s − 0.608·21-s − 0.571·22-s − 1.31·23-s + 0.692·24-s + 1.61·25-s + 0.674·26-s + 3.60·27-s − 0.155·28-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  =  0
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $9.434666624$
$L(\frac12)$  $\approx$  $9.434666624$
$L(\frac{3}{2})$   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 \( 1 - 3.39T + 3T^{2} \)
5 \( 1 - 3.61T + 5T^{2} \)
7 \( 1 + 0.821T + 7T^{2} \)
11 \( 1 + 2.67T + 11T^{2} \)
13 \( 1 - 3.43T + 13T^{2} \)
17 \( 1 + 5.28T + 17T^{2} \)
23 \( 1 + 6.30T + 23T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 - 2.89T + 31T^{2} \)
37 \( 1 - 5.36T + 37T^{2} \)
41 \( 1 + 1.79T + 41T^{2} \)
43 \( 1 + 11.1T + 43T^{2} \)
47 \( 1 - 5.22T + 47T^{2} \)
53 \( 1 - 1.96T + 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 + 9.91T + 61T^{2} \)
67 \( 1 + 9.04T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 - 3.46T + 73T^{2} \)
79 \( 1 + 17.6T + 79T^{2} \)
83 \( 1 + 9.03T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 5.34T + 97T^{2} \)
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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

−7.976217958109292908287246148670, −7.07248193926247400105736169965, −6.34665208767504874112141405184, −5.98344898945205159867450741885, −4.67061763197780637884237971232, −4.36231090197619770317009623439, −3.14849973555261524270187821741, −2.80830411387348317483027476128, −2.03477180091774373592725536646, −1.48238758032561397707754680718, 1.48238758032561397707754680718, 2.03477180091774373592725536646, 2.80830411387348317483027476128, 3.14849973555261524270187821741, 4.36231090197619770317009623439, 4.67061763197780637884237971232, 5.98344898945205159867450741885, 6.34665208767504874112141405184, 7.07248193926247400105736169965, 7.976217958109292908287246148670

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