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  − 2.68·2-s + 5.22·4-s − 0.687·5-s − 3.46·7-s − 8.66·8-s + 1.84·10-s + 4.59·11-s − 5.14·13-s + 9.30·14-s + 12.8·16-s + 5.97·17-s + 2.27·19-s − 3.59·20-s − 12.3·22-s + 23-s − 4.52·25-s + 13.8·26-s − 18.0·28-s + 29-s + 1.71·31-s − 17.1·32-s − 16.0·34-s + 2.38·35-s + 1.76·37-s − 6.12·38-s + 5.95·40-s + 0.217·41-s + ⋯
L(s)  = 1  − 1.90·2-s + 2.61·4-s − 0.307·5-s − 1.30·7-s − 3.06·8-s + 0.584·10-s + 1.38·11-s − 1.42·13-s + 2.48·14-s + 3.21·16-s + 1.44·17-s + 0.523·19-s − 0.803·20-s − 2.63·22-s + 0.208·23-s − 0.905·25-s + 2.71·26-s − 3.41·28-s + 0.185·29-s + 0.308·31-s − 3.03·32-s − 2.75·34-s + 0.402·35-s + 0.289·37-s − 0.994·38-s + 0.942·40-s + 0.0339·41-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 \( 1 + 2.68T + 2T^{2} \)
5 \( 1 + 0.687T + 5T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 - 4.59T + 11T^{2} \)
13 \( 1 + 5.14T + 13T^{2} \)
17 \( 1 - 5.97T + 17T^{2} \)
19 \( 1 - 2.27T + 19T^{2} \)
31 \( 1 - 1.71T + 31T^{2} \)
37 \( 1 - 1.76T + 37T^{2} \)
41 \( 1 - 0.217T + 41T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + 3.23T + 47T^{2} \)
53 \( 1 + 9.45T + 53T^{2} \)
59 \( 1 + 4.74T + 59T^{2} \)
61 \( 1 - 0.257T + 61T^{2} \)
67 \( 1 + 12.3T + 67T^{2} \)
71 \( 1 - 4.10T + 71T^{2} \)
73 \( 1 - 3.50T + 73T^{2} \)
79 \( 1 - 12.8T + 79T^{2} \)
83 \( 1 - 5.50T + 83T^{2} \)
89 \( 1 - 15.5T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−7.61689319890910860045118920675, −7.40858066279374112402330307374, −6.41767204234274100787708147243, −6.22474849543249464440016630215, −4.97290703172238463320706493325, −3.52179565131606956412439701294, −3.11130040070652379805163996801, −2.00119601509055529292394868354, −0.998680315493724787531339416015, 0, 0.998680315493724787531339416015, 2.00119601509055529292394868354, 3.11130040070652379805163996801, 3.52179565131606956412439701294, 4.97290703172238463320706493325, 6.22474849543249464440016630215, 6.41767204234274100787708147243, 7.40858066279374112402330307374, 7.61689319890910860045118920675

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