Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 13^{2} \cdot 167 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 2·5-s + 6-s + 4·7-s + 8-s + 9-s − 2·10-s + 4·11-s + 12-s + 4·14-s − 2·15-s + 16-s − 4·17-s + 18-s + 4·19-s − 2·20-s + 4·21-s + 4·22-s − 4·23-s + 24-s − 25-s + 27-s + 4·28-s − 2·29-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 1.06·14-s − 0.516·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.872·21-s + 0.852·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.755·28-s − 0.371·29-s − 0.365·30-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 169338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 169338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(169338\)    =    \(2 \cdot 3 \cdot 13^{2} \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{169338} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 169338,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;13,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;13,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
13 \( 1 \)
167 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−13.63275482282086, −13.14849052486531, −12.41887647859581, −11.83622256087671, −11.66597249717267, −11.35969341119053, −10.88195743648832, −10.10202064928214, −9.721922089068195, −8.937614184145578, −8.617905496359423, −8.081459040511053, −7.712613348985627, −7.142475182101579, −6.827740786684023, −6.045376454792541, −5.502896457185890, −4.906241689577692, −4.278609808832583, −4.120639792074880, −3.579270693606607, −2.906413031425797, −2.154649779991340, −1.631636137714785, −1.146074932217013, 0, 1.146074932217013, 1.631636137714785, 2.154649779991340, 2.906413031425797, 3.579270693606607, 4.120639792074880, 4.278609808832583, 4.906241689577692, 5.502896457185890, 6.045376454792541, 6.827740786684023, 7.142475182101579, 7.712613348985627, 8.081459040511053, 8.617905496359423, 8.937614184145578, 9.721922089068195, 10.10202064928214, 10.88195743648832, 11.35969341119053, 11.66597249717267, 11.83622256087671, 12.41887647859581, 13.14849052486531, 13.63275482282086

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