Properties

Degree 2
Conductor $ 3 \cdot 61 \cdot 137 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 3

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 2·5-s + 2·6-s − 4·7-s + 9-s + 4·10-s − 6·11-s − 2·12-s − 4·13-s + 8·14-s + 2·15-s − 4·16-s − 6·17-s − 2·18-s − 5·19-s − 4·20-s + 4·21-s + 12·22-s − 7·23-s − 25-s + 8·26-s − 27-s − 8·28-s − 3·29-s − 4·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 0.894·5-s + 0.816·6-s − 1.51·7-s + 1/3·9-s + 1.26·10-s − 1.80·11-s − 0.577·12-s − 1.10·13-s + 2.13·14-s + 0.516·15-s − 16-s − 1.45·17-s − 0.471·18-s − 1.14·19-s − 0.894·20-s + 0.872·21-s + 2.55·22-s − 1.45·23-s − 1/5·25-s + 1.56·26-s − 0.192·27-s − 1.51·28-s − 0.557·29-s − 0.730·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 25071 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 25071 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(25071\)    =    \(3 \cdot 61 \cdot 137\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{25071} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  3
Selberg data  =  $(2,\ 25071,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;61,\;137\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;61,\;137\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
61 \( 1 - T \)
137 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 11 T + p T^{2} \)
73 \( 1 - 17 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−16.18501446085806, −15.75406351389765, −15.45641615454712, −15.07884537596419, −13.71549846974440, −13.46906970243140, −12.78425933460704, −12.31022506523484, −11.80276650617475, −11.05870739838374, −10.45453445615764, −10.25923580265600, −9.751529165039881, −9.037672206771741, −8.426465404939810, −7.974125768445757, −7.267022050480415, −6.947274451576671, −6.314615871584723, −5.524618447814622, −4.723130401893151, −4.159961221209172, −3.285541220496252, −2.389420157701567, −1.901003939307167, 0, 0, 0, 1.901003939307167, 2.389420157701567, 3.285541220496252, 4.159961221209172, 4.723130401893151, 5.524618447814622, 6.314615871584723, 6.947274451576671, 7.267022050480415, 7.974125768445757, 8.426465404939810, 9.037672206771741, 9.751529165039881, 10.25923580265600, 10.45453445615764, 11.05870739838374, 11.80276650617475, 12.31022506523484, 12.78425933460704, 13.46906970243140, 13.71549846974440, 15.07884537596419, 15.45641615454712, 15.75406351389765, 16.18501446085806

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