Properties

Degree 2
Conductor $ 3^{2} \cdot 23 \cdot 29 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.424·2-s − 1.81·4-s − 2.43·5-s + 3.41·7-s + 1.62·8-s + 1.03·10-s + 5.84·11-s − 6.73·13-s − 1.44·14-s + 2.95·16-s − 4.57·17-s + 3.36·19-s + 4.43·20-s − 2.47·22-s + 23-s + 0.941·25-s + 2.85·26-s − 6.21·28-s + 29-s − 2.25·31-s − 4.49·32-s + 1.94·34-s − 8.31·35-s − 4.84·37-s − 1.42·38-s − 3.95·40-s − 4.80·41-s + ⋯
L(s)  = 1  − 0.300·2-s − 0.909·4-s − 1.09·5-s + 1.28·7-s + 0.573·8-s + 0.327·10-s + 1.76·11-s − 1.86·13-s − 0.387·14-s + 0.737·16-s − 1.11·17-s + 0.772·19-s + 0.991·20-s − 0.528·22-s + 0.208·23-s + 0.188·25-s + 0.560·26-s − 1.17·28-s + 0.185·29-s − 0.405·31-s − 0.794·32-s + 0.333·34-s − 1.40·35-s − 0.795·37-s − 0.231·38-s − 0.624·40-s − 0.749·41-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6003} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6003,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;23,\;29\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 \)
23 \( 1 - T \)
29 \( 1 - T \)
good2 \( 1 + 0.424T + 2T^{2} \)
5 \( 1 + 2.43T + 5T^{2} \)
7 \( 1 - 3.41T + 7T^{2} \)
11 \( 1 - 5.84T + 11T^{2} \)
13 \( 1 + 6.73T + 13T^{2} \)
17 \( 1 + 4.57T + 17T^{2} \)
19 \( 1 - 3.36T + 19T^{2} \)
31 \( 1 + 2.25T + 31T^{2} \)
37 \( 1 + 4.84T + 37T^{2} \)
41 \( 1 + 4.80T + 41T^{2} \)
43 \( 1 - 7.09T + 43T^{2} \)
47 \( 1 + 3.50T + 47T^{2} \)
53 \( 1 - 0.803T + 53T^{2} \)
59 \( 1 - 8.37T + 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 - 7.03T + 67T^{2} \)
71 \( 1 - 6.87T + 71T^{2} \)
73 \( 1 + 3.29T + 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 - 15.2T + 83T^{2} \)
89 \( 1 - 7.50T + 89T^{2} \)
97 \( 1 + 9.65T + 97T^{2} \)
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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

−7.58103126963053495623089718295, −7.43891644633323896670493019618, −6.49685966093975582505189105308, −5.19188015720157450385657099917, −4.76736571551160874618515819064, −4.16815919814970937500666980039, −3.51522438320754069010472917218, −2.12865920550031589955437544791, −1.14561483121253766794721599419, 0, 1.14561483121253766794721599419, 2.12865920550031589955437544791, 3.51522438320754069010472917218, 4.16815919814970937500666980039, 4.76736571551160874618515819064, 5.19188015720157450385657099917, 6.49685966093975582505189105308, 7.43891644633323896670493019618, 7.58103126963053495623089718295

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