Properties

Degree 2
Conductor $ 3 \cdot 17 \cdot 157 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.38·2-s − 3-s + 3.69·4-s − 1.70·5-s − 2.38·6-s + 3.54·7-s + 4.03·8-s + 9-s − 4.06·10-s + 1.77·11-s − 3.69·12-s − 3.57·13-s + 8.46·14-s + 1.70·15-s + 2.23·16-s − 17-s + 2.38·18-s − 2.97·19-s − 6.29·20-s − 3.54·21-s + 4.22·22-s − 9.54·23-s − 4.03·24-s − 2.09·25-s − 8.51·26-s − 27-s + 13.0·28-s + ⋯
L(s)  = 1  + 1.68·2-s − 0.577·3-s + 1.84·4-s − 0.762·5-s − 0.973·6-s + 1.34·7-s + 1.42·8-s + 0.333·9-s − 1.28·10-s + 0.534·11-s − 1.06·12-s − 0.990·13-s + 2.26·14-s + 0.440·15-s + 0.559·16-s − 0.242·17-s + 0.562·18-s − 0.682·19-s − 1.40·20-s − 0.774·21-s + 0.901·22-s − 1.98·23-s − 0.823·24-s − 0.418·25-s − 1.67·26-s − 0.192·27-s + 2.47·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8007\)    =    \(3 \cdot 17 \cdot 157\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8007} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8007,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;17,\;157\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;17,\;157\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 + T \)
157 \( 1 + T \)
good2 \( 1 - 2.38T + 2T^{2} \)
5 \( 1 + 1.70T + 5T^{2} \)
7 \( 1 - 3.54T + 7T^{2} \)
11 \( 1 - 1.77T + 11T^{2} \)
13 \( 1 + 3.57T + 13T^{2} \)
19 \( 1 + 2.97T + 19T^{2} \)
23 \( 1 + 9.54T + 23T^{2} \)
29 \( 1 + 2.09T + 29T^{2} \)
31 \( 1 - 3.69T + 31T^{2} \)
37 \( 1 + 3.61T + 37T^{2} \)
41 \( 1 + 1.21T + 41T^{2} \)
43 \( 1 + 2.45T + 43T^{2} \)
47 \( 1 + 5.08T + 47T^{2} \)
53 \( 1 - 3.39T + 53T^{2} \)
59 \( 1 + 1.11T + 59T^{2} \)
61 \( 1 - 4.38T + 61T^{2} \)
67 \( 1 + 7.67T + 67T^{2} \)
71 \( 1 - 1.07T + 71T^{2} \)
73 \( 1 + 8.02T + 73T^{2} \)
79 \( 1 - 0.700T + 79T^{2} \)
83 \( 1 - 8.64T + 83T^{2} \)
89 \( 1 - 8.82T + 89T^{2} \)
97 \( 1 + 19.2T + 97T^{2} \)
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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

−7.29754332698160097305148852702, −6.58314520510186685782749077051, −5.90346832482534560990263877515, −5.22932519896108652024538682831, −4.51477324958363420036867224725, −4.24319370281517188938788524024, −3.49592866243890353851462631709, −2.28589459291503461483901830659, −1.69912838452194452806315818279, 0, 1.69912838452194452806315818279, 2.28589459291503461483901830659, 3.49592866243890353851462631709, 4.24319370281517188938788524024, 4.51477324958363420036867224725, 5.22932519896108652024538682831, 5.90346832482534560990263877515, 6.58314520510186685782749077051, 7.29754332698160097305148852702

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