Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 7^{2} \cdot 13 \cdot 41 $
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 + 5-s − 6-s + 8-s + 9-s + 10-s − 2·11-s − 12-s + 13-s − 15-s + 16-s − 4·17-s + 18-s + 8·19-s + 20-s − 2·22-s + 3·23-s − 24-s − 4·25-s + 26-s − 27-s + 2·29-s − 30-s − 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s − 0.426·22-s + 0.625·23-s − 0.204·24-s − 4/5·25-s + 0.196·26-s − 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.718·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(156702\)    =    \(2 \cdot 3 \cdot 7^{2} \cdot 13 \cdot 41\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{156702} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 156702,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7,\;13,\;41\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;13,\;41\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 \)
13 \( 1 - T \)
41 \( 1 - T \)
good5 \( 1 - T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−13.41267711729321, −13.05349710579624, −12.81079673840111, −12.04545647203710, −11.59776425954709, −11.30560990709901, −10.84033358840274, −10.23345656852583, −9.806812711567372, −9.369403908803415, −8.744136056318461, −8.110714861713692, −7.487212019071804, −7.204225813950927, −6.480809552810071, −6.133048110243061, −5.572729075155499, −4.998660893495371, −4.870625853807235, −3.996439153904761, −3.465768780085529, −2.884761426052154, −2.243867968733130, −1.597064684683571, −0.9360402924608345, 0, 0.9360402924608345, 1.597064684683571, 2.243867968733130, 2.884761426052154, 3.465768780085529, 3.996439153904761, 4.870625853807235, 4.998660893495371, 5.572729075155499, 6.133048110243061, 6.480809552810071, 7.204225813950927, 7.487212019071804, 8.110714861713692, 8.744136056318461, 9.369403908803415, 9.806812711567372, 10.23345656852583, 10.84033358840274, 11.30560990709901, 11.59776425954709, 12.04545647203710, 12.81079673840111, 13.05349710579624, 13.41267711729321

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