Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 7 \cdot 17^{2} $
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 + 4-s − 7-s + 8-s − 4·13-s − 14-s + 16-s + 2·19-s − 5·25-s − 4·26-s − 28-s − 6·29-s + 4·31-s + 32-s − 2·37-s + 2·38-s + 6·41-s + 8·43-s + 12·47-s + 49-s − 5·50-s − 4·52-s − 6·53-s − 56-s − 6·58-s + 6·59-s − 8·61-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 1.10·13-s − 0.267·14-s + 1/4·16-s + 0.458·19-s − 25-s − 0.784·26-s − 0.188·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.328·37-s + 0.324·38-s + 0.937·41-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.707·50-s − 0.554·52-s − 0.824·53-s − 0.133·56-s − 0.787·58-s + 0.781·59-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36414\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{36414} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 36414,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7,\;17\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + T \)
17 \( 1 \)
good5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.17601664100559, −14.47844859879651, −14.18047806567816, −13.61863798064241, −13.03309306058973, −12.62269562705199, −11.95069416683785, −11.78379245736229, −10.94639447066188, −10.50474946164267, −9.833353346709284, −9.389790360869301, −8.859511568478735, −7.907629404223798, −7.451217087148949, −7.141968869530587, −6.189727080666658, −5.862769210348140, −5.217833156312704, −4.545596047004619, −3.996402292642852, −3.336994439626033, −2.573363544953531, −2.116135102159284, −1.064275639059438, 0, 1.064275639059438, 2.116135102159284, 2.573363544953531, 3.336994439626033, 3.996402292642852, 4.545596047004619, 5.217833156312704, 5.862769210348140, 6.189727080666658, 7.141968869530587, 7.451217087148949, 7.907629404223798, 8.859511568478735, 9.389790360869301, 9.833353346709284, 10.50474946164267, 10.94639447066188, 11.78379245736229, 11.95069416683785, 12.62269562705199, 13.03309306058973, 13.61863798064241, 14.18047806567816, 14.47844859879651, 15.17601664100559

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