Properties

Degree $2$
Conductor $92697$
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 + 3·8-s + 9-s − 2·10-s + 11-s − 12-s − 2·13-s − 4·14-s + 2·15-s − 16-s − 2·17-s − 18-s − 2·20-s + 4·21-s − 22-s − 8·23-s + 3·24-s − 25-s + 2·26-s + 27-s − 4·28-s − 6·29-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 + 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s + 0.516·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.447·20-s + 0.872·21-s − 0.213·22-s − 1.66·23-s + 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 92697 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92697 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(92697\)    =    \(3 \cdot 11 \cdot 53^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{92697} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 92697,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
11 \( 1 - T \)
53 \( 1 \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.14999240933527, −13.64804031441972, −13.41631352327258, −12.58453306567459, −12.16631896538218, −11.48701573982526, −11.01951231811669, −10.47365798993375, −9.934434128907541, −9.547391509494804, −9.205443558266444, −8.486864288742694, −8.201456471280061, −7.674239847638478, −7.360940839191766, −6.446549958386704, −5.898115745274285, −5.258914886107317, −4.744853370531480, −4.154935624650716, −3.852692649367752, −2.558424177451645, −2.213891094519340, −1.586064505052433, −1.082237336172454, 0, 1.082237336172454, 1.586064505052433, 2.213891094519340, 2.558424177451645, 3.852692649367752, 4.154935624650716, 4.744853370531480, 5.258914886107317, 5.898115745274285, 6.446549958386704, 7.360940839191766, 7.674239847638478, 8.201456471280061, 8.486864288742694, 9.205443558266444, 9.547391509494804, 9.934434128907541, 10.47365798993375, 11.01951231811669, 11.48701573982526, 12.16631896538218, 12.58453306567459, 13.41631352327258, 13.64804031441972, 14.14999240933527

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