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  − 1.42·2-s + 3-s + 0.0305·4-s + 0.348·5-s − 1.42·6-s − 0.489·7-s + 2.80·8-s + 9-s − 0.497·10-s + 2.33·11-s + 0.0305·12-s + 5.30·13-s + 0.697·14-s + 0.348·15-s − 4.06·16-s − 17-s − 1.42·18-s − 4.47·19-s + 0.0106·20-s − 0.489·21-s − 3.32·22-s + 2.78·23-s + 2.80·24-s − 4.87·25-s − 7.55·26-s + 27-s − 0.0149·28-s + ⋯
L(s)  = 1  − 1.00·2-s + 0.577·3-s + 0.0152·4-s + 0.156·5-s − 0.581·6-s − 0.184·7-s + 0.992·8-s + 0.333·9-s − 0.157·10-s + 0.704·11-s + 0.00881·12-s + 1.47·13-s + 0.186·14-s + 0.0900·15-s − 1.01·16-s − 0.242·17-s − 0.335·18-s − 1.02·19-s + 0.00238·20-s − 0.106·21-s − 0.709·22-s + 0.580·23-s + 0.572·24-s − 0.975·25-s − 1.48·26-s + 0.192·27-s − 0.00282·28-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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 - T \)
17 \( 1 + T \)
157 \( 1 - T \)
good2 \( 1 + 1.42T + 2T^{2} \)
5 \( 1 - 0.348T + 5T^{2} \)
7 \( 1 + 0.489T + 7T^{2} \)
11 \( 1 - 2.33T + 11T^{2} \)
13 \( 1 - 5.30T + 13T^{2} \)
19 \( 1 + 4.47T + 19T^{2} \)
23 \( 1 - 2.78T + 23T^{2} \)
29 \( 1 - 0.169T + 29T^{2} \)
31 \( 1 - 2.78T + 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 - 0.741T + 41T^{2} \)
43 \( 1 + 3.38T + 43T^{2} \)
47 \( 1 + 0.461T + 47T^{2} \)
53 \( 1 + 9.58T + 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 - 1.32T + 61T^{2} \)
67 \( 1 + 2.14T + 67T^{2} \)
71 \( 1 - 8.64T + 71T^{2} \)
73 \( 1 - 1.89T + 73T^{2} \)
79 \( 1 + 5.45T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + 6.89T + 89T^{2} \)
97 \( 1 - 7.97T + 97T^{2} \)
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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

−7.83781372372055007854194971316, −6.75544517518391862411037682248, −6.50331284740611745870413560355, −5.45269827691523190114233848180, −4.45118557041684091203259147460, −3.88801289764690603179719708301, −3.08550523867298736512464386779, −1.82265883838740572447078489876, −1.35101860963026278176394700064, 0, 1.35101860963026278176394700064, 1.82265883838740572447078489876, 3.08550523867298736512464386779, 3.88801289764690603179719708301, 4.45118557041684091203259147460, 5.45269827691523190114233848180, 6.50331284740611745870413560355, 6.75544517518391862411037682248, 7.83781372372055007854194971316

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