Properties

Label 2-84-1.1-c3-0-0
Degree $2$
Conductor $84$
Sign $1$
Analytic cond. $4.95616$
Root an. cond. $2.22624$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 6·5-s + 7·7-s + 9·9-s + 36·11-s + 62·13-s − 18·15-s + 114·17-s − 76·19-s − 21·21-s − 24·23-s − 89·25-s − 27·27-s + 54·29-s − 112·31-s − 108·33-s + 42·35-s − 178·37-s − 186·39-s + 378·41-s − 172·43-s + 54·45-s − 192·47-s + 49·49-s − 342·51-s − 402·53-s + 216·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.536·5-s + 0.377·7-s + 1/3·9-s + 0.986·11-s + 1.32·13-s − 0.309·15-s + 1.62·17-s − 0.917·19-s − 0.218·21-s − 0.217·23-s − 0.711·25-s − 0.192·27-s + 0.345·29-s − 0.648·31-s − 0.569·33-s + 0.202·35-s − 0.790·37-s − 0.763·39-s + 1.43·41-s − 0.609·43-s + 0.178·45-s − 0.595·47-s + 1/7·49-s − 0.939·51-s − 1.04·53-s + 0.529·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(84\)    =    \(2^{2} \cdot 3 \cdot 7\)
Sign: $1$
Analytic conductor: \(4.95616\)
Root analytic conductor: \(2.22624\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{84} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 84,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.527469426\)
\(L(\frac12)\) \(\approx\) \(1.527469426\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
7 \( 1 - p T \)
good5 \( 1 - 6 T + p^{3} T^{2} \)
11 \( 1 - 36 T + p^{3} T^{2} \)
13 \( 1 - 62 T + p^{3} T^{2} \)
17 \( 1 - 114 T + p^{3} T^{2} \)
19 \( 1 + 4 p T + p^{3} T^{2} \)
23 \( 1 + 24 T + p^{3} T^{2} \)
29 \( 1 - 54 T + p^{3} T^{2} \)
31 \( 1 + 112 T + p^{3} T^{2} \)
37 \( 1 + 178 T + p^{3} T^{2} \)
41 \( 1 - 378 T + p^{3} T^{2} \)
43 \( 1 + 4 p T + p^{3} T^{2} \)
47 \( 1 + 192 T + p^{3} T^{2} \)
53 \( 1 + 402 T + p^{3} T^{2} \)
59 \( 1 - 396 T + p^{3} T^{2} \)
61 \( 1 - 254 T + p^{3} T^{2} \)
67 \( 1 + 1012 T + p^{3} T^{2} \)
71 \( 1 - 840 T + p^{3} T^{2} \)
73 \( 1 - 890 T + p^{3} T^{2} \)
79 \( 1 - 80 T + p^{3} T^{2} \)
83 \( 1 + 108 T + p^{3} T^{2} \)
89 \( 1 + 1638 T + p^{3} T^{2} \)
97 \( 1 - 1010 T + p^{3} T^{2} \)
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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

−13.83201726133467512318490119763, −12.61856591791949487538252939790, −11.58908163730096665968715601934, −10.58315257361744133740154201358, −9.426228672939652439237526663022, −8.114389789415304142426981918414, −6.51779561773031791860008209629, −5.56030199737424878272255152376, −3.86728510554865626200271519258, −1.42982332922689640035017855530, 1.42982332922689640035017855530, 3.86728510554865626200271519258, 5.56030199737424878272255152376, 6.51779561773031791860008209629, 8.114389789415304142426981918414, 9.426228672939652439237526663022, 10.58315257361744133740154201358, 11.58908163730096665968715601934, 12.61856591791949487538252939790, 13.83201726133467512318490119763

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