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 − 2.78·3-s + 4-s − 2.51·5-s − 2.78·6-s − 0.381·7-s + 8-s + 4.78·9-s − 2.51·10-s + 1.85·11-s − 2.78·12-s − 1.51·13-s − 0.381·14-s + 7.01·15-s + 16-s − 3.53·17-s + 4.78·18-s − 19-s − 2.51·20-s + 1.06·21-s + 1.85·22-s + 9.05·23-s − 2.78·24-s + 1.32·25-s − 1.51·26-s − 4.97·27-s − 0.381·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.61·3-s + 0.5·4-s − 1.12·5-s − 1.13·6-s − 0.144·7-s + 0.353·8-s + 1.59·9-s − 0.795·10-s + 0.558·11-s − 0.805·12-s − 0.419·13-s − 0.101·14-s + 1.81·15-s + 0.250·16-s − 0.856·17-s + 1.12·18-s − 0.229·19-s − 0.562·20-s + 0.232·21-s + 0.395·22-s + 1.88·23-s − 0.569·24-s + 0.264·25-s − 0.296·26-s − 0.957·27-s − 0.0720·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9283533448$
$L(\frac12)$  $\approx$  $0.9283533448$
$L(\frac{3}{2})$   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 \( 1 + 2.78T + 3T^{2} \)
5 \( 1 + 2.51T + 5T^{2} \)
7 \( 1 + 0.381T + 7T^{2} \)
11 \( 1 - 1.85T + 11T^{2} \)
13 \( 1 + 1.51T + 13T^{2} \)
17 \( 1 + 3.53T + 17T^{2} \)
23 \( 1 - 9.05T + 23T^{2} \)
29 \( 1 - 5.58T + 29T^{2} \)
31 \( 1 + 6.76T + 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 - 0.614T + 41T^{2} \)
43 \( 1 + 5.81T + 43T^{2} \)
47 \( 1 - 10.1T + 47T^{2} \)
53 \( 1 - 6.61T + 53T^{2} \)
59 \( 1 + 7.56T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 7.71T + 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 3.95T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 + 4.88T + 89T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−7.28951341729879272912975399438, −7.03176980072608837724901252115, −6.49044541324282297431395375507, −5.60893715110972748120361445183, −5.01596737755505683319414313825, −4.44677718512729903480936359373, −3.80424443165713998977857095807, −2.91489892882541488603905753987, −1.58723180350206969169945983206, −0.47445746990977074752704130490, 0.47445746990977074752704130490, 1.58723180350206969169945983206, 2.91489892882541488603905753987, 3.80424443165713998977857095807, 4.44677718512729903480936359373, 5.01596737755505683319414313825, 5.60893715110972748120361445183, 6.49044541324282297431395375507, 7.03176980072608837724901252115, 7.28951341729879272912975399438

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