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Properties

Degree $2$
Conductor $126$
Sign $0.635 - 0.771i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (1 + 1.41i)3-s + (−0.499 + 0.866i)4-s + (−1.72 + 2.98i)5-s + (0.724 − 1.57i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−1.00 + 2.82i)9-s + 3.44·10-s + (−1 − 1.73i)11-s + (−1.72 + 0.158i)12-s + (2.44 − 4.24i)13-s + (0.499 − 0.866i)14-s + (−5.94 + 0.548i)15-s + (−0.5 − 0.866i)16-s + 2·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.577 + 0.816i)3-s + (−0.249 + 0.433i)4-s + (−0.771 + 1.33i)5-s + (0.295 − 0.642i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.333 + 0.942i)9-s + 1.09·10-s + (−0.301 − 0.522i)11-s + (−0.497 + 0.0458i)12-s + (0.679 − 1.17i)13-s + (0.133 − 0.231i)14-s + (−1.53 + 0.141i)15-s + (−0.125 − 0.216i)16-s + 0.485·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.635 - 0.771i$
Motivic weight: \(1\)
Character: $\chi_{126} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.635 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859205 + 0.405491i\)
\(L(\frac12)\) \(\approx\) \(0.859205 + 0.405491i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1 - 1.41i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (1.72 - 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.44 + 4.24i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 7.44T + 19T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.44 + 2.51i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3 - 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 7.79T + 37T^{2} \)
41 \( 1 + (-4.89 + 8.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.44 - 2.51i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.89 - 8.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.10T + 53T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.72 + 9.91i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.55 + 2.68i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.89T + 71T^{2} \)
73 \( 1 - 2.89T + 73T^{2} \)
79 \( 1 + (3.94 + 6.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.10T + 89T^{2} \)
97 \( 1 + (-3.44 - 5.97i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.74424642319222629701536003275, −12.23148477541442390070865912028, −10.97913546905467841987557907386, −10.70353069980969479585778177130, −9.491111726959563972177591639996, −8.233924276768020124611546630617, −7.47418572570541941636517090938, −5.46345688450125332645148071140, −3.57209220922600236818641414030, −2.95395167591308361300504164103, 1.28371195834214731635635577479, 3.96152791550059179482244910907, 5.40559281042754067551752899743, 7.08531890079274455098805681232, 7.82524293043590474923847189075, 8.781712046802330974465872367532, 9.566166560568983836561329147333, 11.47398549566425395104320339983, 12.31380147742070860493937868146, 13.36591250760106181581103952932

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