Properties

Degree $2$
Conductor $8619$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s + 2·5-s − 6-s − 3·8-s + 9-s + 2·10-s − 4·11-s + 12-s − 2·15-s − 16-s + 17-s + 18-s + 4·19-s − 2·20-s − 4·22-s + 3·24-s − 25-s − 27-s − 2·29-s − 2·30-s + 8·31-s + 5·32-s + 4·33-s + 34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s + 0.288·12-s − 0.516·15-s − 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.852·22-s + 0.612·24-s − 1/5·25-s − 0.192·27-s − 0.371·29-s − 0.365·30-s + 1.43·31-s + 0.883·32-s + 0.696·33-s + 0.171·34-s − 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8619\)    =    \(3 \cdot 13^{2} \cdot 17\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8619} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8619,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.777683275\)
\(L(\frac12)\) \(\approx\) \(1.777683275\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−17.09382302872005, −16.05995893002252, −15.81771422494014, −15.03938993467861, −14.39684562346518, −13.84107844717104, −13.28294194217416, −13.00396112794137, −12.30475399854331, −11.66548474035753, −11.07400482972314, −10.14414287234233, −9.815984116822178, −9.333564567457869, −8.239359100446947, −7.899563502118070, −6.771340089450234, −6.202203009592251, −5.501026375820312, −5.113625075244482, −4.541552878128231, −3.473766095181401, −2.834061403222773, −1.817155309338313, −0.5958994507119832, 0.5958994507119832, 1.817155309338313, 2.834061403222773, 3.473766095181401, 4.541552878128231, 5.113625075244482, 5.501026375820312, 6.202203009592251, 6.771340089450234, 7.899563502118070, 8.239359100446947, 9.333564567457869, 9.815984116822178, 10.14414287234233, 11.07400482972314, 11.66548474035753, 12.30475399854331, 13.00396112794137, 13.28294194217416, 13.84107844717104, 14.39684562346518, 15.03938993467861, 15.81771422494014, 16.05995893002252, 17.09382302872005

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