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  − 0.857·2-s − 3-s − 1.26·4-s + 0.241·5-s + 0.857·6-s − 3.40·7-s + 2.79·8-s + 9-s − 0.207·10-s + 4.52·11-s + 1.26·12-s − 3.66·13-s + 2.92·14-s − 0.241·15-s + 0.130·16-s − 17-s − 0.857·18-s + 3.02·19-s − 0.305·20-s + 3.40·21-s − 3.87·22-s − 6.34·23-s − 2.79·24-s − 4.94·25-s + 3.14·26-s − 27-s + 4.31·28-s + ⋯
L(s)  = 1  − 0.606·2-s − 0.577·3-s − 0.632·4-s + 0.108·5-s + 0.349·6-s − 1.28·7-s + 0.989·8-s + 0.333·9-s − 0.0655·10-s + 1.36·11-s + 0.365·12-s − 1.01·13-s + 0.781·14-s − 0.0624·15-s + 0.0327·16-s − 0.242·17-s − 0.202·18-s + 0.693·19-s − 0.0684·20-s + 0.743·21-s − 0.827·22-s − 1.32·23-s − 0.571·24-s − 0.988·25-s + 0.616·26-s − 0.192·27-s + 0.815·28-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 \( 1 + 0.857T + 2T^{2} \)
5 \( 1 - 0.241T + 5T^{2} \)
7 \( 1 + 3.40T + 7T^{2} \)
11 \( 1 - 4.52T + 11T^{2} \)
13 \( 1 + 3.66T + 13T^{2} \)
19 \( 1 - 3.02T + 19T^{2} \)
23 \( 1 + 6.34T + 23T^{2} \)
29 \( 1 - 5.43T + 29T^{2} \)
31 \( 1 - 2.05T + 31T^{2} \)
37 \( 1 - 9.39T + 37T^{2} \)
41 \( 1 - 0.671T + 41T^{2} \)
43 \( 1 + 6.04T + 43T^{2} \)
47 \( 1 + 0.151T + 47T^{2} \)
53 \( 1 + 8.36T + 53T^{2} \)
59 \( 1 - 1.97T + 59T^{2} \)
61 \( 1 - 0.358T + 61T^{2} \)
67 \( 1 - 1.12T + 67T^{2} \)
71 \( 1 + 4.67T + 71T^{2} \)
73 \( 1 - 0.743T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 - 4.57T + 83T^{2} \)
89 \( 1 - 1.89T + 89T^{2} \)
97 \( 1 + 1.97T + 97T^{2} \)
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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

−7.59022500808131176005093506865, −6.63400245831892718473920237417, −6.32993506075207654060025766127, −5.47562055374689058783825836277, −4.56771285376727127094059095938, −4.03227713477740032442800553465, −3.19373442266329157372028994520, −2.00397163400179779676860573235, −0.914792416765416847664497004550, 0, 0.914792416765416847664497004550, 2.00397163400179779676860573235, 3.19373442266329157372028994520, 4.03227713477740032442800553465, 4.56771285376727127094059095938, 5.47562055374689058783825836277, 6.32993506075207654060025766127, 6.63400245831892718473920237417, 7.59022500808131176005093506865

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