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  + 0.289·2-s − 1.91·4-s − 0.894·5-s + 3.31·7-s − 1.13·8-s − 0.258·10-s − 3.47·11-s + 1.33·13-s + 0.957·14-s + 3.50·16-s + 4.10·17-s − 2.37·19-s + 1.71·20-s − 1.00·22-s + 23-s − 4.20·25-s + 0.385·26-s − 6.34·28-s + 29-s − 8.16·31-s + 3.27·32-s + 1.18·34-s − 2.96·35-s − 7.23·37-s − 0.686·38-s + 1.01·40-s + 2.74·41-s + ⋯
L(s)  = 1  + 0.204·2-s − 0.958·4-s − 0.399·5-s + 1.25·7-s − 0.400·8-s − 0.0817·10-s − 1.04·11-s + 0.370·13-s + 0.255·14-s + 0.876·16-s + 0.995·17-s − 0.544·19-s + 0.383·20-s − 0.214·22-s + 0.208·23-s − 0.840·25-s + 0.0756·26-s − 1.19·28-s + 0.185·29-s − 1.46·31-s + 0.579·32-s + 0.203·34-s − 0.500·35-s − 1.18·37-s − 0.111·38-s + 0.160·40-s + 0.428·41-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 \( 1 - 0.289T + 2T^{2} \)
5 \( 1 + 0.894T + 5T^{2} \)
7 \( 1 - 3.31T + 7T^{2} \)
11 \( 1 + 3.47T + 11T^{2} \)
13 \( 1 - 1.33T + 13T^{2} \)
17 \( 1 - 4.10T + 17T^{2} \)
19 \( 1 + 2.37T + 19T^{2} \)
31 \( 1 + 8.16T + 31T^{2} \)
37 \( 1 + 7.23T + 37T^{2} \)
41 \( 1 - 2.74T + 41T^{2} \)
43 \( 1 - 1.38T + 43T^{2} \)
47 \( 1 - 2.36T + 47T^{2} \)
53 \( 1 - 1.52T + 53T^{2} \)
59 \( 1 - 15.2T + 59T^{2} \)
61 \( 1 - 4.80T + 61T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + 0.717T + 71T^{2} \)
73 \( 1 + 11.3T + 73T^{2} \)
79 \( 1 - 8.33T + 79T^{2} \)
83 \( 1 + 0.376T + 83T^{2} \)
89 \( 1 - 17.2T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−7.73402600994961633808686349435, −7.37493936639894499955252505862, −6.03503521158660557078169441642, −5.33575450129885613346788753804, −4.96206401075285416280076106091, −4.03799857025117016791334914952, −3.50389248625558640976919419362, −2.32326871620683720800583674605, −1.24424332575653202718190008361, 0, 1.24424332575653202718190008361, 2.32326871620683720800583674605, 3.50389248625558640976919419362, 4.03799857025117016791334914952, 4.96206401075285416280076106091, 5.33575450129885613346788753804, 6.03503521158660557078169441642, 7.37493936639894499955252505862, 7.73402600994961633808686349435

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