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Properties

Label 2-8712-1.1-c1-0-74
Degree $2$
Conductor $8712$
Sign $-1$
Analytic cond. $69.5656$
Root an. cond. $8.34060$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 4·7-s − 4·13-s + 4·17-s + 4·19-s + 3·23-s − 4·25-s + 8·29-s + 9·31-s + 4·35-s − 5·37-s − 12·41-s − 8·43-s − 4·47-s + 9·49-s + 10·53-s − 7·59-s + 8·61-s + 4·65-s + 11·67-s + 9·71-s − 4·73-s − 8·79-s − 4·85-s + 89-s + 16·91-s − 4·95-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.51·7-s − 1.10·13-s + 0.970·17-s + 0.917·19-s + 0.625·23-s − 4/5·25-s + 1.48·29-s + 1.61·31-s + 0.676·35-s − 0.821·37-s − 1.87·41-s − 1.21·43-s − 0.583·47-s + 9/7·49-s + 1.37·53-s − 0.911·59-s + 1.02·61-s + 0.496·65-s + 1.34·67-s + 1.06·71-s − 0.468·73-s − 0.900·79-s − 0.433·85-s + 0.105·89-s + 1.67·91-s − 0.410·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8712\)    =    \(2^{3} \cdot 3^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(69.5656\)
Root analytic conductor: \(8.34060\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8712,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 - 9 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 7 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 11 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 - T + p T^{2} \)
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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

−7.23992186489116086704773996175, −6.84524178880070604308751397568, −6.17524641343519814707054492541, −5.25238383153064554587868794729, −4.74061870618693709969334768405, −3.56683574885690052757458683174, −3.22770744843772426622181097807, −2.42659888175239657323854236606, −1.03509263435064377927401135065, 0, 1.03509263435064377927401135065, 2.42659888175239657323854236606, 3.22770744843772426622181097807, 3.56683574885690052757458683174, 4.74061870618693709969334768405, 5.25238383153064554587868794729, 6.17524641343519814707054492541, 6.84524178880070604308751397568, 7.23992186489116086704773996175

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