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  + 1.05·2-s − 0.895·4-s + 3.10·5-s + 3.07·7-s − 3.04·8-s + 3.26·10-s − 3.20·11-s − 0.0429·13-s + 3.23·14-s − 1.40·16-s − 4.01·17-s − 7.31·19-s − 2.78·20-s − 3.36·22-s + 23-s + 4.64·25-s − 0.0451·26-s − 2.75·28-s + 29-s + 0.714·31-s + 4.60·32-s − 4.21·34-s + 9.54·35-s − 6.91·37-s − 7.69·38-s − 9.45·40-s − 5.11·41-s + ⋯
L(s)  = 1  + 0.743·2-s − 0.447·4-s + 1.38·5-s + 1.16·7-s − 1.07·8-s + 1.03·10-s − 0.966·11-s − 0.0119·13-s + 0.863·14-s − 0.351·16-s − 0.973·17-s − 1.67·19-s − 0.622·20-s − 0.718·22-s + 0.208·23-s + 0.929·25-s − 0.00886·26-s − 0.520·28-s + 0.185·29-s + 0.128·31-s + 0.814·32-s − 0.723·34-s + 1.61·35-s − 1.13·37-s − 1.24·38-s − 1.49·40-s − 0.798·41-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 \( 1 - 1.05T + 2T^{2} \)
5 \( 1 - 3.10T + 5T^{2} \)
7 \( 1 - 3.07T + 7T^{2} \)
11 \( 1 + 3.20T + 11T^{2} \)
13 \( 1 + 0.0429T + 13T^{2} \)
17 \( 1 + 4.01T + 17T^{2} \)
19 \( 1 + 7.31T + 19T^{2} \)
31 \( 1 - 0.714T + 31T^{2} \)
37 \( 1 + 6.91T + 37T^{2} \)
41 \( 1 + 5.11T + 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 + 5.42T + 47T^{2} \)
53 \( 1 - 2.84T + 53T^{2} \)
59 \( 1 + 10.8T + 59T^{2} \)
61 \( 1 + 8.48T + 61T^{2} \)
67 \( 1 - 3.06T + 67T^{2} \)
71 \( 1 + 3.65T + 71T^{2} \)
73 \( 1 - 9.89T + 73T^{2} \)
79 \( 1 - 3.31T + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 - 6.29T + 89T^{2} \)
97 \( 1 + 0.736T + 97T^{2} \)
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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

−7.88483701924668387308643866846, −6.60311326202085015639543421589, −6.28258966192661626116729009045, −5.21950963370609818652880457686, −5.03042150364851648216372866463, −4.34148677984455583680498353757, −3.23451568950067941900573892156, −2.25006252306489354720311119192, −1.71303345871422803056652458446, 0, 1.71303345871422803056652458446, 2.25006252306489354720311119192, 3.23451568950067941900573892156, 4.34148677984455583680498353757, 5.03042150364851648216372866463, 5.21950963370609818652880457686, 6.28258966192661626116729009045, 6.60311326202085015639543421589, 7.88483701924668387308643866846

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