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  − 1.96·2-s + 1.84·4-s − 2.84·5-s − 2.21·7-s + 0.305·8-s + 5.57·10-s + 1.29·11-s − 0.673·13-s + 4.34·14-s − 4.28·16-s + 1.19·17-s − 5.61·19-s − 5.24·20-s − 2.54·22-s + 23-s + 3.07·25-s + 1.32·26-s − 4.08·28-s + 29-s − 4.05·31-s + 7.79·32-s − 2.33·34-s + 6.29·35-s + 3.39·37-s + 11.0·38-s − 0.867·40-s + 9.19·41-s + ⋯
L(s)  = 1  − 1.38·2-s + 0.922·4-s − 1.27·5-s − 0.837·7-s + 0.107·8-s + 1.76·10-s + 0.390·11-s − 0.186·13-s + 1.16·14-s − 1.07·16-s + 0.289·17-s − 1.28·19-s − 1.17·20-s − 0.542·22-s + 0.208·23-s + 0.614·25-s + 0.259·26-s − 0.771·28-s + 0.185·29-s − 0.728·31-s + 1.37·32-s − 0.401·34-s + 1.06·35-s + 0.557·37-s + 1.78·38-s − 0.137·40-s + 1.43·41-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 \( 1 + 1.96T + 2T^{2} \)
5 \( 1 + 2.84T + 5T^{2} \)
7 \( 1 + 2.21T + 7T^{2} \)
11 \( 1 - 1.29T + 11T^{2} \)
13 \( 1 + 0.673T + 13T^{2} \)
17 \( 1 - 1.19T + 17T^{2} \)
19 \( 1 + 5.61T + 19T^{2} \)
31 \( 1 + 4.05T + 31T^{2} \)
37 \( 1 - 3.39T + 37T^{2} \)
41 \( 1 - 9.19T + 41T^{2} \)
43 \( 1 - 9.58T + 43T^{2} \)
47 \( 1 + 4.53T + 47T^{2} \)
53 \( 1 - 3.31T + 53T^{2} \)
59 \( 1 - 7.04T + 59T^{2} \)
61 \( 1 + 13.0T + 61T^{2} \)
67 \( 1 - 4.14T + 67T^{2} \)
71 \( 1 + 4.40T + 71T^{2} \)
73 \( 1 + 6.12T + 73T^{2} \)
79 \( 1 - 1.11T + 79T^{2} \)
83 \( 1 + 2.93T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 - 0.608T + 97T^{2} \)
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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

−7.77123881221988626054548975267, −7.33389524340374074391613576008, −6.61761669436921476110093940956, −5.88446645814106297942655438109, −4.54932419837864365820321401339, −4.06281076510238537235980520494, −3.13373952182232621054014554023, −2.12172605422210244851273045718, −0.857962473595032970441260868976, 0, 0.857962473595032970441260868976, 2.12172605422210244851273045718, 3.13373952182232621054014554023, 4.06281076510238537235980520494, 4.54932419837864365820321401339, 5.88446645814106297942655438109, 6.61761669436921476110093940956, 7.33389524340374074391613576008, 7.77123881221988626054548975267

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