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 + 1.01·3-s + 4-s + 3.08·5-s + 1.01·6-s + 1.61·7-s + 8-s − 1.96·9-s + 3.08·10-s − 1.92·11-s + 1.01·12-s − 5.39·13-s + 1.61·14-s + 3.13·15-s + 16-s + 6.81·17-s − 1.96·18-s − 19-s + 3.08·20-s + 1.64·21-s − 1.92·22-s + 2.24·23-s + 1.01·24-s + 4.52·25-s − 5.39·26-s − 5.04·27-s + 1.61·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.586·3-s + 0.5·4-s + 1.38·5-s + 0.414·6-s + 0.610·7-s + 0.353·8-s − 0.656·9-s + 0.976·10-s − 0.581·11-s + 0.293·12-s − 1.49·13-s + 0.431·14-s + 0.809·15-s + 0.250·16-s + 1.65·17-s − 0.464·18-s − 0.229·19-s + 0.690·20-s + 0.357·21-s − 0.410·22-s + 0.467·23-s + 0.207·24-s + 0.905·25-s − 1.05·26-s − 0.970·27-s + 0.305·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $5.528928412$
$L(\frac12)$  $\approx$  $5.528928412$
$L(\frac{3}{2})$   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 \( 1 - 1.01T + 3T^{2} \)
5 \( 1 - 3.08T + 5T^{2} \)
7 \( 1 - 1.61T + 7T^{2} \)
11 \( 1 + 1.92T + 11T^{2} \)
13 \( 1 + 5.39T + 13T^{2} \)
17 \( 1 - 6.81T + 17T^{2} \)
23 \( 1 - 2.24T + 23T^{2} \)
29 \( 1 - 7.76T + 29T^{2} \)
31 \( 1 - 0.778T + 31T^{2} \)
37 \( 1 - 3.25T + 37T^{2} \)
41 \( 1 - 8.93T + 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + 1.95T + 47T^{2} \)
53 \( 1 - 8.08T + 53T^{2} \)
59 \( 1 + 14.1T + 59T^{2} \)
61 \( 1 - 4.28T + 61T^{2} \)
67 \( 1 - 4.68T + 67T^{2} \)
71 \( 1 + 2.72T + 71T^{2} \)
73 \( 1 - 0.822T + 73T^{2} \)
79 \( 1 - 6.10T + 79T^{2} \)
83 \( 1 + 10.9T + 83T^{2} \)
89 \( 1 - 0.942T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−7.73174231962433179546338549621, −7.25488032381132375894255558030, −6.12531250624975189776908224073, −5.73014670451682645281904998227, −5.06083911010923043096962319526, −4.51462719789300287041725028143, −3.25400531141056940492720809973, −2.59012284607199419273169006975, −2.21309007011915453579468368472, −1.04551105144370405978280025518, 1.04551105144370405978280025518, 2.21309007011915453579468368472, 2.59012284607199419273169006975, 3.25400531141056940492720809973, 4.51462719789300287041725028143, 5.06083911010923043096962319526, 5.73014670451682645281904998227, 6.12531250624975189776908224073, 7.25488032381132375894255558030, 7.73174231962433179546338549621

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