Properties

Label 2-126-1.1-c9-0-18
Degree $2$
Conductor $126$
Sign $-1$
Analytic cond. $64.8945$
Root an. cond. $8.05571$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s + 256·4-s − 560·5-s − 2.40e3·7-s + 4.09e3·8-s − 8.96e3·10-s + 5.41e4·11-s − 1.13e5·13-s − 3.84e4·14-s + 6.55e4·16-s − 6.26e3·17-s + 2.57e5·19-s − 1.43e5·20-s + 8.66e5·22-s + 2.66e5·23-s − 1.63e6·25-s − 1.81e6·26-s − 6.14e5·28-s − 1.57e6·29-s − 4.63e6·31-s + 1.04e6·32-s − 1.00e5·34-s + 1.34e6·35-s − 1.19e7·37-s + 4.11e6·38-s − 2.29e6·40-s − 2.19e7·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.400·5-s − 0.377·7-s + 0.353·8-s − 0.283·10-s + 1.11·11-s − 1.09·13-s − 0.267·14-s + 1/4·16-s − 0.0181·17-s + 0.452·19-s − 0.200·20-s + 0.788·22-s + 0.198·23-s − 0.839·25-s − 0.777·26-s − 0.188·28-s − 0.413·29-s − 0.901·31-s + 0.176·32-s − 0.0128·34-s + 0.151·35-s − 1.04·37-s + 0.320·38-s − 0.141·40-s − 1.21·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(64.8945\)
Root analytic conductor: \(8.05571\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: $\chi_{126} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 126,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{4} T \)
3 \( 1 \)
7 \( 1 + p^{4} T \)
good5 \( 1 + 112 p T + p^{9} T^{2} \)
11 \( 1 - 54152 T + p^{9} T^{2} \)
13 \( 1 + 113172 T + p^{9} T^{2} \)
17 \( 1 + 6262 T + p^{9} T^{2} \)
19 \( 1 - 257078 T + p^{9} T^{2} \)
23 \( 1 - 266000 T + p^{9} T^{2} \)
29 \( 1 + 1574714 T + p^{9} T^{2} \)
31 \( 1 + 4637484 T + p^{9} T^{2} \)
37 \( 1 + 11946238 T + p^{9} T^{2} \)
41 \( 1 + 21909126 T + p^{9} T^{2} \)
43 \( 1 - 27520592 T + p^{9} T^{2} \)
47 \( 1 + 52927836 T + p^{9} T^{2} \)
53 \( 1 + 16221222 T + p^{9} T^{2} \)
59 \( 1 - 140509618 T + p^{9} T^{2} \)
61 \( 1 + 202963560 T + p^{9} T^{2} \)
67 \( 1 - 153734572 T + p^{9} T^{2} \)
71 \( 1 + 3938816 p T + p^{9} T^{2} \)
73 \( 1 + 404022830 T + p^{9} T^{2} \)
79 \( 1 + 130689816 T + p^{9} T^{2} \)
83 \( 1 + 420134014 T + p^{9} T^{2} \)
89 \( 1 - 469542390 T + p^{9} T^{2} \)
97 \( 1 + 872501690 T + p^{9} T^{2} \)
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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

−11.47054034527794040227470232224, −10.11156900857991763742397154309, −9.074154130643214154326199889161, −7.58431591176391000653690152941, −6.70359884408964945921411774564, −5.43745817917469330891292559993, −4.19941663698617374818084472573, −3.18099304120056605470306551768, −1.69388304820034955467579563413, 0, 1.69388304820034955467579563413, 3.18099304120056605470306551768, 4.19941663698617374818084472573, 5.43745817917469330891292559993, 6.70359884408964945921411774564, 7.58431591176391000653690152941, 9.074154130643214154326199889161, 10.11156900857991763742397154309, 11.47054034527794040227470232224

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