Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5 \cdot 167 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s − 2·11-s + 12-s + 14-s + 15-s + 16-s + 4·17-s + 18-s − 19-s + 20-s + 21-s − 2·22-s + 3·23-s + 24-s + 25-s + 27-s + 28-s + 2·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.229·19-s + 0.223·20-s + 0.218·21-s − 0.426·22-s + 0.625·23-s + 0.204·24-s + 1/5·25-s + 0.192·27-s + 0.188·28-s + 0.371·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 5010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 5010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(5010\)    =    \(2 \cdot 3 \cdot 5 \cdot 167\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{5010} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 5010,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.704449998$
$L(\frac12)$  $\approx$  $4.704449998$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
167 \( 1 + T \)
good7 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 5 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−17.84056394915007, −17.00650934694560, −16.42641937706416, −15.84588304690171, −15.09194738478841, −14.57831140679372, −14.23807533910442, −13.35014613098647, −13.05838493781232, −12.41343527366860, −11.55980436419913, −11.05176350249090, −10.12437287244595, −9.797636065358821, −8.864063084810147, −7.990144245718622, −7.718429510787997, −6.634361553292071, −6.108757461449092, −5.068516772788162, −4.746726787338541, −3.615626479439204, −2.920098672357400, −2.142490710263313, −1.106926601600278, 1.106926601600278, 2.142490710263313, 2.920098672357400, 3.615626479439204, 4.746726787338541, 5.068516772788162, 6.108757461449092, 6.634361553292071, 7.718429510787997, 7.990144245718622, 8.864063084810147, 9.797636065358821, 10.12437287244595, 11.05176350249090, 11.55980436419913, 12.41343527366860, 13.05838493781232, 13.35014613098647, 14.23807533910442, 14.57831140679372, 15.09194738478841, 15.84588304690171, 16.42641937706416, 17.00650934694560, 17.84056394915007

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