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 + 2.25·3-s + 4-s + 0.0902·5-s + 2.25·6-s + 2.22·7-s + 8-s + 2.10·9-s + 0.0902·10-s + 2.60·11-s + 2.25·12-s + 2.02·13-s + 2.22·14-s + 0.203·15-s + 16-s + 3.50·17-s + 2.10·18-s − 19-s + 0.0902·20-s + 5.02·21-s + 2.60·22-s + 5.88·23-s + 2.25·24-s − 4.99·25-s + 2.02·26-s − 2.02·27-s + 2.22·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.30·3-s + 0.5·4-s + 0.0403·5-s + 0.922·6-s + 0.840·7-s + 0.353·8-s + 0.701·9-s + 0.0285·10-s + 0.786·11-s + 0.652·12-s + 0.562·13-s + 0.594·14-s + 0.0526·15-s + 0.250·16-s + 0.850·17-s + 0.496·18-s − 0.229·19-s + 0.0201·20-s + 1.09·21-s + 0.555·22-s + 1.22·23-s + 0.461·24-s − 0.998·25-s + 0.397·26-s − 0.389·27-s + 0.420·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $6.863791402$
$L(\frac12)$  $\approx$  $6.863791402$
$L(\frac{3}{2})$   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 \( 1 - 2.25T + 3T^{2} \)
5 \( 1 - 0.0902T + 5T^{2} \)
7 \( 1 - 2.22T + 7T^{2} \)
11 \( 1 - 2.60T + 11T^{2} \)
13 \( 1 - 2.02T + 13T^{2} \)
17 \( 1 - 3.50T + 17T^{2} \)
23 \( 1 - 5.88T + 23T^{2} \)
29 \( 1 - 7.15T + 29T^{2} \)
31 \( 1 + 7.83T + 31T^{2} \)
37 \( 1 + 1.37T + 37T^{2} \)
41 \( 1 - 3.84T + 41T^{2} \)
43 \( 1 + 9.72T + 43T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + 4.98T + 53T^{2} \)
59 \( 1 - 2.42T + 59T^{2} \)
61 \( 1 + 4.40T + 61T^{2} \)
67 \( 1 + 5.90T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 8.97T + 73T^{2} \)
79 \( 1 + 7.84T + 79T^{2} \)
83 \( 1 + 17.5T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + 2.03T + 97T^{2} \)
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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

−7.85680254784560145386573381280, −7.25393249305802021233778988867, −6.47355219909584135519788596568, −5.65016451552315820675632998589, −4.93602252771994405969136317542, −4.07453650427385491801479267031, −3.53672703355347482587721631840, −2.83964737170911965397457076431, −1.91609484639753098889272608045, −1.23594010764837262098698367780, 1.23594010764837262098698367780, 1.91609484639753098889272608045, 2.83964737170911965397457076431, 3.53672703355347482587721631840, 4.07453650427385491801479267031, 4.93602252771994405969136317542, 5.65016451552315820675632998589, 6.47355219909584135519788596568, 7.25393249305802021233778988867, 7.85680254784560145386573381280

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