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 + 3.14·3-s + 4-s + 1.08·5-s + 3.14·6-s − 1.59·7-s + 8-s + 6.92·9-s + 1.08·10-s + 5.80·11-s + 3.14·12-s + 2.89·13-s − 1.59·14-s + 3.41·15-s + 16-s + 3.40·17-s + 6.92·18-s − 19-s + 1.08·20-s − 5.03·21-s + 5.80·22-s + 2.00·23-s + 3.14·24-s − 3.82·25-s + 2.89·26-s + 12.3·27-s − 1.59·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.81·3-s + 0.5·4-s + 0.484·5-s + 1.28·6-s − 0.604·7-s + 0.353·8-s + 2.30·9-s + 0.342·10-s + 1.75·11-s + 0.909·12-s + 0.803·13-s − 0.427·14-s + 0.880·15-s + 0.250·16-s + 0.825·17-s + 1.63·18-s − 0.229·19-s + 0.242·20-s − 1.09·21-s + 1.23·22-s + 0.417·23-s + 0.642·24-s − 0.765·25-s + 0.568·26-s + 2.37·27-s − 0.302·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.290737953$
$L(\frac12)$  $\approx$  $8.290737953$
$L(\frac{3}{2})$   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 \( 1 - 3.14T + 3T^{2} \)
5 \( 1 - 1.08T + 5T^{2} \)
7 \( 1 + 1.59T + 7T^{2} \)
11 \( 1 - 5.80T + 11T^{2} \)
13 \( 1 - 2.89T + 13T^{2} \)
17 \( 1 - 3.40T + 17T^{2} \)
23 \( 1 - 2.00T + 23T^{2} \)
29 \( 1 + 3.53T + 29T^{2} \)
31 \( 1 + 3.83T + 31T^{2} \)
37 \( 1 - 3.57T + 37T^{2} \)
41 \( 1 + 4.86T + 41T^{2} \)
43 \( 1 + 7.60T + 43T^{2} \)
47 \( 1 + 7.58T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 8.16T + 59T^{2} \)
61 \( 1 + 4.47T + 61T^{2} \)
67 \( 1 - 14.6T + 67T^{2} \)
71 \( 1 + 9.60T + 71T^{2} \)
73 \( 1 + 8.36T + 73T^{2} \)
79 \( 1 - 1.39T + 79T^{2} \)
83 \( 1 - 4.62T + 83T^{2} \)
89 \( 1 + 1.72T + 89T^{2} \)
97 \( 1 + 18.1T + 97T^{2} \)
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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

−7.898941175812781832282927864029, −7.05273289109702920314003659619, −6.52948047969500401538016626506, −5.89977286365789278465431806965, −4.80573969725337167887236354353, −3.83443503302919644592968444384, −3.58423973585966244531872876205, −2.95378065611608403025426500936, −1.82483396257356583818491532127, −1.42562161583496808465957919754, 1.42562161583496808465957919754, 1.82483396257356583818491532127, 2.95378065611608403025426500936, 3.58423973585966244531872876205, 3.83443503302919644592968444384, 4.80573969725337167887236354353, 5.89977286365789278465431806965, 6.52948047969500401538016626506, 7.05273289109702920314003659619, 7.898941175812781832282927864029

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