Properties

Degree 2
Conductor $ 2 \cdot 19 \cdot 211 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.44·3-s + 4-s − 1.97·5-s + 2.44·6-s − 1.80·7-s + 8-s + 2.97·9-s − 1.97·10-s + 0.774·11-s + 2.44·12-s − 4.35·13-s − 1.80·14-s − 4.83·15-s + 16-s − 1.48·17-s + 2.97·18-s + 19-s − 1.97·20-s − 4.41·21-s + 0.774·22-s − 1.62·23-s + 2.44·24-s − 1.08·25-s − 4.35·26-s − 0.0536·27-s − 1.80·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.41·3-s + 0.5·4-s − 0.885·5-s + 0.998·6-s − 0.682·7-s + 0.353·8-s + 0.992·9-s − 0.625·10-s + 0.233·11-s + 0.705·12-s − 1.20·13-s − 0.482·14-s − 1.24·15-s + 0.250·16-s − 0.359·17-s + 0.701·18-s + 0.229·19-s − 0.442·20-s − 0.963·21-s + 0.165·22-s − 0.339·23-s + 0.499·24-s − 0.216·25-s − 0.854·26-s − 0.0103·27-s − 0.341·28-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  =  1
Selberg data  =  $(2,\ 8018,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - 2.44T + 3T^{2} \)
5 \( 1 + 1.97T + 5T^{2} \)
7 \( 1 + 1.80T + 7T^{2} \)
11 \( 1 - 0.774T + 11T^{2} \)
13 \( 1 + 4.35T + 13T^{2} \)
17 \( 1 + 1.48T + 17T^{2} \)
23 \( 1 + 1.62T + 23T^{2} \)
29 \( 1 - 5.23T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + 7.04T + 37T^{2} \)
41 \( 1 + 0.380T + 41T^{2} \)
43 \( 1 + 5.09T + 43T^{2} \)
47 \( 1 + 5.06T + 47T^{2} \)
53 \( 1 + 2.03T + 53T^{2} \)
59 \( 1 + 5.98T + 59T^{2} \)
61 \( 1 + 6.74T + 61T^{2} \)
67 \( 1 - 6.96T + 67T^{2} \)
71 \( 1 + 6.40T + 71T^{2} \)
73 \( 1 + 5.49T + 73T^{2} \)
79 \( 1 - 10.6T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 - 9.67T + 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−7.56746708298463994853690309741, −6.83670289943717669277499137884, −6.31629816299279394291577267917, −5.12428666647134122597401963192, −4.44524853166979538979428747413, −3.81591266731287334517016043617, −3.03374618406614967387905254658, −2.69879521732599959270286193877, −1.63425692909703502647080694375, 0, 1.63425692909703502647080694375, 2.69879521732599959270286193877, 3.03374618406614967387905254658, 3.81591266731287334517016043617, 4.44524853166979538979428747413, 5.12428666647134122597401963192, 6.31629816299279394291577267917, 6.83670289943717669277499137884, 7.56746708298463994853690309741

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