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  − 0.206·2-s − 3-s − 1.95·4-s − 2.96·5-s + 0.206·6-s − 3.29·7-s + 0.817·8-s + 9-s + 0.612·10-s − 2.02·11-s + 1.95·12-s − 4.52·13-s + 0.681·14-s + 2.96·15-s + 3.74·16-s − 17-s − 0.206·18-s + 2.84·19-s + 5.80·20-s + 3.29·21-s + 0.418·22-s − 6.15·23-s − 0.817·24-s + 3.78·25-s + 0.934·26-s − 27-s + 6.45·28-s + ⋯
L(s)  = 1  − 0.146·2-s − 0.577·3-s − 0.978·4-s − 1.32·5-s + 0.0843·6-s − 1.24·7-s + 0.289·8-s + 0.333·9-s + 0.193·10-s − 0.610·11-s + 0.565·12-s − 1.25·13-s + 0.182·14-s + 0.765·15-s + 0.936·16-s − 0.242·17-s − 0.0486·18-s + 0.652·19-s + 1.29·20-s + 0.719·21-s + 0.0891·22-s − 1.28·23-s − 0.166·24-s + 0.756·25-s + 0.183·26-s − 0.192·27-s + 1.22·28-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 \( 1 + 0.206T + 2T^{2} \)
5 \( 1 + 2.96T + 5T^{2} \)
7 \( 1 + 3.29T + 7T^{2} \)
11 \( 1 + 2.02T + 11T^{2} \)
13 \( 1 + 4.52T + 13T^{2} \)
19 \( 1 - 2.84T + 19T^{2} \)
23 \( 1 + 6.15T + 23T^{2} \)
29 \( 1 - 3.35T + 29T^{2} \)
31 \( 1 + 3.97T + 31T^{2} \)
37 \( 1 + 6.02T + 37T^{2} \)
41 \( 1 + 5.50T + 41T^{2} \)
43 \( 1 + 0.805T + 43T^{2} \)
47 \( 1 - 2.71T + 47T^{2} \)
53 \( 1 - 7.81T + 53T^{2} \)
59 \( 1 - 0.000491T + 59T^{2} \)
61 \( 1 - 10.2T + 61T^{2} \)
67 \( 1 - 6.98T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + 0.595T + 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 - 6.31T + 89T^{2} \)
97 \( 1 - 6.62T + 97T^{2} \)
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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

−7.43156076838393935455433424963, −7.01628130270502532444519494384, −6.06746664416621412425357017963, −5.20759011004790631830966313010, −4.75935896549892568167714562514, −3.77579269172065013440246429594, −3.49135520164297972296728108657, −2.29761636093617758020672457654, −0.61253215798925192097417670556, 0, 0.61253215798925192097417670556, 2.29761636093617758020672457654, 3.49135520164297972296728108657, 3.77579269172065013440246429594, 4.75935896549892568167714562514, 5.20759011004790631830966313010, 6.06746664416621412425357017963, 7.01628130270502532444519494384, 7.43156076838393935455433424963

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