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  − 1.54·2-s − 3-s + 0.392·4-s + 3.59·5-s + 1.54·6-s − 3.93·7-s + 2.48·8-s + 9-s − 5.55·10-s + 3.85·11-s − 0.392·12-s − 1.41·13-s + 6.08·14-s − 3.59·15-s − 4.63·16-s − 17-s − 1.54·18-s − 0.145·19-s + 1.40·20-s + 3.93·21-s − 5.96·22-s − 0.400·23-s − 2.48·24-s + 7.89·25-s + 2.19·26-s − 27-s − 1.54·28-s + ⋯
L(s)  = 1  − 1.09·2-s − 0.577·3-s + 0.196·4-s + 1.60·5-s + 0.631·6-s − 1.48·7-s + 0.879·8-s + 0.333·9-s − 1.75·10-s + 1.16·11-s − 0.113·12-s − 0.393·13-s + 1.62·14-s − 0.927·15-s − 1.15·16-s − 0.242·17-s − 0.364·18-s − 0.0334·19-s + 0.314·20-s + 0.859·21-s − 1.27·22-s − 0.0835·23-s − 0.507·24-s + 1.57·25-s + 0.430·26-s − 0.192·27-s − 0.291·28-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 \( 1 + 1.54T + 2T^{2} \)
5 \( 1 - 3.59T + 5T^{2} \)
7 \( 1 + 3.93T + 7T^{2} \)
11 \( 1 - 3.85T + 11T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
19 \( 1 + 0.145T + 19T^{2} \)
23 \( 1 + 0.400T + 23T^{2} \)
29 \( 1 + 4.77T + 29T^{2} \)
31 \( 1 - 9.59T + 31T^{2} \)
37 \( 1 + 8.11T + 37T^{2} \)
41 \( 1 - 2.99T + 41T^{2} \)
43 \( 1 - 3.02T + 43T^{2} \)
47 \( 1 + 8.05T + 47T^{2} \)
53 \( 1 - 4.69T + 53T^{2} \)
59 \( 1 + 8.98T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 + 5.49T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 - 6.38T + 73T^{2} \)
79 \( 1 - 8.76T + 79T^{2} \)
83 \( 1 + 6.68T + 83T^{2} \)
89 \( 1 + 7.17T + 89T^{2} \)
97 \( 1 + 11.2T + 97T^{2} \)
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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

−7.37111061201146515709336399796, −6.57287184098402268886742729629, −6.43909187387008055911434753816, −5.62435933649499824284154960250, −4.80329228291280260513482407726, −3.91672648730122424102293378486, −2.85336300688698370698262449294, −1.88762568286239313033461501873, −1.12664855102117093871172896802, 0, 1.12664855102117093871172896802, 1.88762568286239313033461501873, 2.85336300688698370698262449294, 3.91672648730122424102293378486, 4.80329228291280260513482407726, 5.62435933649499824284154960250, 6.43909187387008055911434753816, 6.57287184098402268886742729629, 7.37111061201146515709336399796

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