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 + 3.04·3-s + 4-s − 3.75·5-s + 3.04·6-s − 3.60·7-s + 8-s + 6.26·9-s − 3.75·10-s + 5.45·11-s + 3.04·12-s + 4.27·13-s − 3.60·14-s − 11.4·15-s + 16-s − 0.0995·17-s + 6.26·18-s − 19-s − 3.75·20-s − 10.9·21-s + 5.45·22-s − 7.43·23-s + 3.04·24-s + 9.06·25-s + 4.27·26-s + 9.93·27-s − 3.60·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.75·3-s + 0.5·4-s − 1.67·5-s + 1.24·6-s − 1.36·7-s + 0.353·8-s + 2.08·9-s − 1.18·10-s + 1.64·11-s + 0.878·12-s + 1.18·13-s − 0.962·14-s − 2.94·15-s + 0.250·16-s − 0.0241·17-s + 1.47·18-s − 0.229·19-s − 0.838·20-s − 2.39·21-s + 1.16·22-s − 1.54·23-s + 0.621·24-s + 1.81·25-s + 0.839·26-s + 1.91·27-s − 0.680·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.735747784$
$L(\frac12)$  $\approx$  $4.735747784$
$L(\frac{3}{2})$   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 \( 1 - 3.04T + 3T^{2} \)
5 \( 1 + 3.75T + 5T^{2} \)
7 \( 1 + 3.60T + 7T^{2} \)
11 \( 1 - 5.45T + 11T^{2} \)
13 \( 1 - 4.27T + 13T^{2} \)
17 \( 1 + 0.0995T + 17T^{2} \)
23 \( 1 + 7.43T + 23T^{2} \)
29 \( 1 - 1.40T + 29T^{2} \)
31 \( 1 - 2.61T + 31T^{2} \)
37 \( 1 + 8.63T + 37T^{2} \)
41 \( 1 - 7.19T + 41T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 - 4.78T + 47T^{2} \)
53 \( 1 - 0.455T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 - 6.96T + 61T^{2} \)
67 \( 1 - 5.68T + 67T^{2} \)
71 \( 1 + 4.11T + 71T^{2} \)
73 \( 1 - 6.68T + 73T^{2} \)
79 \( 1 - 6.05T + 79T^{2} \)
83 \( 1 + 3.24T + 83T^{2} \)
89 \( 1 - 4.65T + 89T^{2} \)
97 \( 1 + 0.313T + 97T^{2} \)
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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

−7.79916768963347622549082869112, −7.23997181356958260472475071474, −6.54288719565384825170642914970, −6.02885855724206166491962123397, −4.33622383236145815336223390151, −4.03900234255458016887993438142, −3.55653694484745135047792536630, −3.14232398210044210164350760077, −2.10536177167588823501861494885, −0.904714775506372420145213549706, 0.904714775506372420145213549706, 2.10536177167588823501861494885, 3.14232398210044210164350760077, 3.55653694484745135047792536630, 4.03900234255458016887993438142, 4.33622383236145815336223390151, 6.02885855724206166491962123397, 6.54288719565384825170642914970, 7.23997181356958260472475071474, 7.79916768963347622549082869112

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