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 − 2.90·3-s + 4-s − 4.15·5-s − 2.90·6-s + 3.02·7-s + 8-s + 5.46·9-s − 4.15·10-s + 1.94·11-s − 2.90·12-s + 7.15·13-s + 3.02·14-s + 12.0·15-s + 16-s + 1.01·17-s + 5.46·18-s − 19-s − 4.15·20-s − 8.78·21-s + 1.94·22-s + 3.81·23-s − 2.90·24-s + 12.2·25-s + 7.15·26-s − 7.17·27-s + 3.02·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.67·3-s + 0.5·4-s − 1.85·5-s − 1.18·6-s + 1.14·7-s + 0.353·8-s + 1.82·9-s − 1.31·10-s + 0.587·11-s − 0.839·12-s + 1.98·13-s + 0.807·14-s + 3.11·15-s + 0.250·16-s + 0.247·17-s + 1.28·18-s − 0.229·19-s − 0.928·20-s − 1.91·21-s + 0.415·22-s + 0.795·23-s − 0.593·24-s + 2.44·25-s + 1.40·26-s − 1.38·27-s + 0.570·28-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  =  $(2,\ 8018,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.030319901$
$L(\frac12)$  $\approx$  $2.030319901$
$L(\frac{3}{2})$   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 \( 1 + 2.90T + 3T^{2} \)
5 \( 1 + 4.15T + 5T^{2} \)
7 \( 1 - 3.02T + 7T^{2} \)
11 \( 1 - 1.94T + 11T^{2} \)
13 \( 1 - 7.15T + 13T^{2} \)
17 \( 1 - 1.01T + 17T^{2} \)
23 \( 1 - 3.81T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 - 4.20T + 31T^{2} \)
37 \( 1 - 9.42T + 37T^{2} \)
41 \( 1 - 0.697T + 41T^{2} \)
43 \( 1 - 7.53T + 43T^{2} \)
47 \( 1 + 5.51T + 47T^{2} \)
53 \( 1 + 3.63T + 53T^{2} \)
59 \( 1 - 7.99T + 59T^{2} \)
61 \( 1 + 1.55T + 61T^{2} \)
67 \( 1 - 1.68T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 - 3.13T + 73T^{2} \)
79 \( 1 + 1.68T + 79T^{2} \)
83 \( 1 + 5.75T + 83T^{2} \)
89 \( 1 + 8.16T + 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−7.77878742197896617820813925929, −6.85700255390219704438537525232, −6.42134693480717038852386822862, −5.73446191070808294004061083080, −4.79688584682475912303670301291, −4.44371303078621444850247155385, −3.93410420732722897855609462121, −3.00629448038752147711813438463, −1.21851025714975922792933211902, −0.881104887502317971773292226972, 0.881104887502317971773292226972, 1.21851025714975922792933211902, 3.00629448038752147711813438463, 3.93410420732722897855609462121, 4.44371303078621444850247155385, 4.79688584682475912303670301291, 5.73446191070808294004061083080, 6.42134693480717038852386822862, 6.85700255390219704438537525232, 7.77878742197896617820813925929

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