Properties

Degree 2
Conductor $ 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  − 2.48·2-s + 0.221·3-s + 4.17·4-s − 2.50·5-s − 0.550·6-s + 2.33·7-s − 5.39·8-s − 2.95·9-s + 6.23·10-s − 4.94·11-s + 0.924·12-s − 3.44·13-s − 5.78·14-s − 0.555·15-s + 5.05·16-s + 4.24·17-s + 7.32·18-s + 19-s − 10.4·20-s + 0.516·21-s + 12.2·22-s − 3.69·23-s − 1.19·24-s + 1.29·25-s + 8.55·26-s − 1.31·27-s + 9.71·28-s + ⋯
L(s)  = 1  − 1.75·2-s + 0.127·3-s + 2.08·4-s − 1.12·5-s − 0.224·6-s + 0.880·7-s − 1.90·8-s − 0.983·9-s + 1.97·10-s − 1.49·11-s + 0.266·12-s − 0.955·13-s − 1.54·14-s − 0.143·15-s + 1.26·16-s + 1.02·17-s + 1.72·18-s + 0.229·19-s − 2.33·20-s + 0.112·21-s + 2.61·22-s − 0.771·23-s − 0.243·24-s + 0.258·25-s + 1.67·26-s − 0.253·27-s + 1.83·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4009\)    =    \(19 \cdot 211\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{4009} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 4009,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{19,\;211\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{19,\;211\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad19 \( 1 - T \)
211 \( 1 - T \)
good2 \( 1 + 2.48T + 2T^{2} \)
3 \( 1 - 0.221T + 3T^{2} \)
5 \( 1 + 2.50T + 5T^{2} \)
7 \( 1 - 2.33T + 7T^{2} \)
11 \( 1 + 4.94T + 11T^{2} \)
13 \( 1 + 3.44T + 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
23 \( 1 + 3.69T + 23T^{2} \)
29 \( 1 - 3.97T + 29T^{2} \)
31 \( 1 - 8.72T + 31T^{2} \)
37 \( 1 - 9.05T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 - 1.63T + 43T^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 2.69T + 59T^{2} \)
61 \( 1 - 14.5T + 61T^{2} \)
67 \( 1 + 0.0512T + 67T^{2} \)
71 \( 1 + 12.5T + 71T^{2} \)
73 \( 1 - 0.953T + 73T^{2} \)
79 \( 1 - 0.539T + 79T^{2} \)
83 \( 1 - 4.69T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 + 17.2T + 97T^{2} \)
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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

−8.145156926677405437259412596419, −7.78991617232036513298706532772, −7.19637448770087144953202116805, −6.02607072344221915185118077237, −5.19241951922625401879598619238, −4.24980434518783803253893330355, −2.80959568329311938252433494731, −2.47715114219567106867810996108, −0.976066840455629509767778549600, 0, 0.976066840455629509767778549600, 2.47715114219567106867810996108, 2.80959568329311938252433494731, 4.24980434518783803253893330355, 5.19241951922625401879598619238, 6.02607072344221915185118077237, 7.19637448770087144953202116805, 7.78991617232036513298706532772, 8.145156926677405437259412596419

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