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 + 2.24·3-s + 4-s + 3.57·5-s + 2.24·6-s + 0.516·7-s + 8-s + 2.02·9-s + 3.57·10-s + 3.00·11-s + 2.24·12-s + 1.63·13-s + 0.516·14-s + 8.00·15-s + 16-s − 0.530·17-s + 2.02·18-s − 19-s + 3.57·20-s + 1.15·21-s + 3.00·22-s − 2.37·23-s + 2.24·24-s + 7.74·25-s + 1.63·26-s − 2.17·27-s + 0.516·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.29·3-s + 0.5·4-s + 1.59·5-s + 0.915·6-s + 0.195·7-s + 0.353·8-s + 0.676·9-s + 1.12·10-s + 0.905·11-s + 0.647·12-s + 0.453·13-s + 0.137·14-s + 2.06·15-s + 0.250·16-s − 0.128·17-s + 0.478·18-s − 0.229·19-s + 0.798·20-s + 0.252·21-s + 0.640·22-s − 0.495·23-s + 0.457·24-s + 1.54·25-s + 0.320·26-s − 0.419·27-s + 0.0975·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.062176491$
$L(\frac12)$  $\approx$  $8.062176491$
$L(\frac{3}{2})$   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 \( 1 - 2.24T + 3T^{2} \)
5 \( 1 - 3.57T + 5T^{2} \)
7 \( 1 - 0.516T + 7T^{2} \)
11 \( 1 - 3.00T + 11T^{2} \)
13 \( 1 - 1.63T + 13T^{2} \)
17 \( 1 + 0.530T + 17T^{2} \)
23 \( 1 + 2.37T + 23T^{2} \)
29 \( 1 - 4.16T + 29T^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 + 8.08T + 37T^{2} \)
41 \( 1 - 2.76T + 41T^{2} \)
43 \( 1 + 3.09T + 43T^{2} \)
47 \( 1 + 6.39T + 47T^{2} \)
53 \( 1 - 6.86T + 53T^{2} \)
59 \( 1 - 7.54T + 59T^{2} \)
61 \( 1 - 13.9T + 61T^{2} \)
67 \( 1 + 7.40T + 67T^{2} \)
71 \( 1 - 9.46T + 71T^{2} \)
73 \( 1 + 5.84T + 73T^{2} \)
79 \( 1 - 14.2T + 79T^{2} \)
83 \( 1 + 8.41T + 83T^{2} \)
89 \( 1 + 5.57T + 89T^{2} \)
97 \( 1 - 9.07T + 97T^{2} \)
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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

−7.952232460579129947365831429690, −6.85824994328556912176502296617, −6.53328366097006615140072318745, −5.68639265790037072191064240026, −5.11430946466640221428434083587, −4.08402752562497529450932920331, −3.49290549290875671237560458206, −2.62637437365548542168524112272, −1.98356176895615122278261085113, −1.38187804476734323521067551434, 1.38187804476734323521067551434, 1.98356176895615122278261085113, 2.62637437365548542168524112272, 3.49290549290875671237560458206, 4.08402752562497529450932920331, 5.11430946466640221428434083587, 5.68639265790037072191064240026, 6.53328366097006615140072318745, 6.85824994328556912176502296617, 7.952232460579129947365831429690

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