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Properties

Label 2-126-63.41-c1-0-7
Degree $2$
Conductor $126$
Sign $0.341 + 0.940i$
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  + (0.866 − 0.5i)2-s + (0.0967 − 1.72i)3-s + (0.499 − 0.866i)4-s + (0.183 − 0.317i)5-s + (−0.780 − 1.54i)6-s + (−0.624 + 2.57i)7-s − 0.999i·8-s + (−2.98 − 0.334i)9-s − 0.366i·10-s + (0.579 − 0.334i)11-s + (−1.44 − 0.948i)12-s + (0.867 + 0.500i)13-s + (0.744 + 2.53i)14-s + (−0.531 − 0.347i)15-s + (−0.5 − 0.866i)16-s + 4.98·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.0558 − 0.998i)3-s + (0.249 − 0.433i)4-s + (0.0819 − 0.141i)5-s + (−0.318 − 0.631i)6-s + (−0.235 + 0.971i)7-s − 0.353i·8-s + (−0.993 − 0.111i)9-s − 0.115i·10-s + (0.174 − 0.100i)11-s + (−0.418 − 0.273i)12-s + (0.240 + 0.138i)13-s + (0.199 + 0.678i)14-s + (−0.137 − 0.0897i)15-s + (−0.125 − 0.216i)16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.341 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.341 + 0.940i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.341 + 0.940i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17990 - 0.826990i\)
\(L(\frac12)\) \(\approx\) \(1.17990 - 0.826990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.0967 + 1.72i)T \)
7 \( 1 + (0.624 - 2.57i)T \)
good5 \( 1 + (-0.183 + 0.317i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.579 + 0.334i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.867 - 0.500i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.98T + 17T^{2} \)
19 \( 1 - 6.35iT - 19T^{2} \)
23 \( 1 + (6.66 + 3.84i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.58 + 0.914i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.47 + 3.16i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.16T + 37T^{2} \)
41 \( 1 + (-2.15 + 3.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.24 - 3.89i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.16 + 7.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (4.36 - 7.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.29 + 2.47i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.44 + 9.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.49iT - 71T^{2} \)
73 \( 1 + 4.07iT - 73T^{2} \)
79 \( 1 + (4.17 + 7.23i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.50 + 14.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + (-14.9 + 8.60i)T + (48.5 - 84.0i)T^{2} \)
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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

−12.93849074476169513012394296982, −12.28003552895765187908726871399, −11.62976331938711579907735386998, −10.17520593567578556389003889564, −8.861071252947852406414693415573, −7.73001675709558753051868500686, −6.23570489643954552346397555555, −5.52050253705868402202212364498, −3.45324970481293391234563155593, −1.88207767779536472002557961304, 3.21838208193317123931231656561, 4.31208636071578698735969313392, 5.55295138246210281316659527495, 6.89683945830747836106050108902, 8.164521258763114127287451048423, 9.547738620388256050669934820749, 10.48771123804329421476108444608, 11.44285607983500434070605102121, 12.69554100934106072271414950338, 13.98016901659877725078825472805

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