Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \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 + 5-s − 7-s + 3·8-s − 10-s − 6·13-s + 14-s − 16-s − 8·19-s − 20-s + 8·23-s + 25-s + 6·26-s + 28-s − 2·29-s − 4·31-s − 5·32-s − 35-s + 2·37-s + 8·38-s + 3·40-s − 6·41-s + 4·43-s − 8·46-s − 8·47-s + 49-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.377·7-s + 1.06·8-s − 0.316·10-s − 1.66·13-s + 0.267·14-s − 1/4·16-s − 1.83·19-s − 0.223·20-s + 1.66·23-s + 1/5·25-s + 1.17·26-s + 0.188·28-s − 0.371·29-s − 0.718·31-s − 0.883·32-s − 0.169·35-s + 0.328·37-s + 1.29·38-s + 0.474·40-s − 0.937·41-s + 0.609·43-s − 1.17·46-s − 1.16·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(91035\)    =    \(3^{2} \cdot 5 \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{91035} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 91035,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;5,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 \)
good2 \( 1 + T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−14.22858927482673, −13.45479199108873, −13.01429506164146, −12.76381684087660, −12.34426098021916, −11.46084060496254, −11.02176476619624, −10.42776152391942, −10.09769366826244, −9.577789237620261, −9.100257599021185, −8.796321473426983, −8.168771027468686, −7.514459232833492, −7.161535862642481, −6.563431571603663, −5.989426154280033, −5.200354730721928, −4.758669002335155, −4.425866140315044, −3.510647918394131, −2.916193653727692, −2.105875044818925, −1.690545022315217, −0.6387756184465831, 0, 0.6387756184465831, 1.690545022315217, 2.105875044818925, 2.916193653727692, 3.510647918394131, 4.425866140315044, 4.758669002335155, 5.200354730721928, 5.989426154280033, 6.563431571603663, 7.161535862642481, 7.514459232833492, 8.168771027468686, 8.796321473426983, 9.100257599021185, 9.577789237620261, 10.09769366826244, 10.42776152391942, 11.02176476619624, 11.46084060496254, 12.34426098021916, 12.76381684087660, 13.01429506164146, 13.45479199108873, 14.22858927482673

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