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Properties

Label 2-126-9.4-c1-0-2
Degree $2$
Conductor $126$
Sign $0.399 - 0.916i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $1$
Arithmetic yes
Rational no
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 + (1.18 + 1.26i)3-s + (−0.499 + 0.866i)4-s + (0.686 − 1.18i)5-s + (−0.5 + 1.65i)6-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.186 + 2.99i)9-s + 1.37·10-s + (−2.18 − 3.78i)11-s + (−1.68 + 0.396i)12-s + (−1 + 1.73i)13-s + (0.499 − 0.866i)14-s + (2.31 − 0.543i)15-s + (−0.5 − 0.866i)16-s − 4.37·17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.684 + 0.728i)3-s + (−0.249 + 0.433i)4-s + (0.306 − 0.531i)5-s + (−0.204 + 0.677i)6-s + (−0.188 − 0.327i)7-s − 0.353·8-s + (−0.0620 + 0.998i)9-s + 0.433·10-s + (−0.659 − 1.14i)11-s + (−0.486 + 0.114i)12-s + (−0.277 + 0.480i)13-s + (0.133 − 0.231i)14-s + (0.597 − 0.140i)15-s + (−0.125 − 0.216i)16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.399 - 0.916i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21916 + 0.798450i\)
\(L(\frac12)\) \(\approx\) \(1.21916 + 0.798450i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.18 - 1.26i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.686 + 1.18i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.18 + 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.37T + 17T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + (-3.68 + 6.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.37 + 2.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (5.18 - 8.98i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.55 + 7.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.74T + 53T^{2} \)
59 \( 1 + (3.55 - 6.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.05 - 12.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + 5.11T + 73T^{2} \)
79 \( 1 + (6.05 + 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.74 - 4.75i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.25T + 89T^{2} \)
97 \( 1 + (4.55 + 7.89i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.52800061268005924137919066798, −13.15673273802718201535620283140, −11.47021211354963289362890518497, −10.30125430188152363037732730294, −9.087993171069103384190256023281, −8.418965819437021315515620696375, −7.07303048009666643407173618507, −5.50179419082868554959466102205, −4.46452392562003441426142369778, −2.98073045426211311221302203718, 2.10732594047403510097966828064, 3.24916221363599159764289039433, 5.15364717351957514406832532988, 6.67740331843246986413126198964, 7.69647613661721236000456706668, 9.185892151627262300803817088652, 9.998694303896251120023691501641, 11.29375485244275901812657898716, 12.41835551712099721948875087964, 13.12866599705885963758926237063

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