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Properties

Label 2-126-7.4-c7-0-12
Degree $2$
Conductor $126$
Sign $0.928 - 0.370i$
Analytic cond. $39.3605$
Root an. cond. $6.27379$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 + 6.92i)2-s + (−31.9 + 55.4i)4-s + (−95.0 − 164. i)5-s + (−515. + 746. i)7-s − 511.·8-s + (760. − 1.31e3i)10-s + (3.12e3 − 5.41e3i)11-s − 4.18e3·13-s + (−7.23e3 − 587. i)14-s + (−2.04e3 − 3.54e3i)16-s + (288. − 499. i)17-s + (7.72e3 + 1.33e4i)19-s + 1.21e4·20-s + 5.00e4·22-s + (2.93e4 + 5.08e4i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.339 − 0.588i)5-s + (−0.568 + 0.822i)7-s − 0.353·8-s + (0.240 − 0.416i)10-s + (0.708 − 1.22i)11-s − 0.528·13-s + (−0.704 − 0.0571i)14-s + (−0.125 − 0.216i)16-s + (0.0142 − 0.0246i)17-s + (0.258 + 0.447i)19-s + 0.339·20-s + 1.00·22-s + (0.502 + 0.870i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.928 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.928 - 0.370i$
Analytic conductor: \(39.3605\)
Root analytic conductor: \(6.27379\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :7/2),\ 0.928 - 0.370i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.986203643\)
\(L(\frac12)\) \(\approx\) \(1.986203643\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 - 6.92i)T \)
3 \( 1 \)
7 \( 1 + (515. - 746. i)T \)
good5 \( 1 + (95.0 + 164. i)T + (-3.90e4 + 6.76e4i)T^{2} \)
11 \( 1 + (-3.12e3 + 5.41e3i)T + (-9.74e6 - 1.68e7i)T^{2} \)
13 \( 1 + 4.18e3T + 6.27e7T^{2} \)
17 \( 1 + (-288. + 499. i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (-7.72e3 - 1.33e4i)T + (-4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-2.93e4 - 5.08e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 - 2.31e4T + 1.72e10T^{2} \)
31 \( 1 + (-1.51e5 + 2.61e5i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 + (-1.11e5 - 1.93e5i)T + (-4.74e10 + 8.22e10i)T^{2} \)
41 \( 1 - 3.30e5T + 1.94e11T^{2} \)
43 \( 1 - 2.84e5T + 2.71e11T^{2} \)
47 \( 1 + (-6.53e5 - 1.13e6i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + (-5.51e5 + 9.54e5i)T + (-5.87e11 - 1.01e12i)T^{2} \)
59 \( 1 + (-1.79e4 + 3.11e4i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-6.02e5 - 1.04e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.11e5 + 1.93e5i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 - 4.83e6T + 9.09e12T^{2} \)
73 \( 1 + (1.74e6 - 3.01e6i)T + (-5.52e12 - 9.56e12i)T^{2} \)
79 \( 1 + (3.72e6 + 6.45e6i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 - 4.70e6T + 2.71e13T^{2} \)
89 \( 1 + (-1.71e6 - 2.97e6i)T + (-2.21e13 + 3.83e13i)T^{2} \)
97 \( 1 - 3.79e6T + 8.07e13T^{2} \)
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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

−12.18129840416613420694728937362, −11.41007776847667659992905151342, −9.646688425377058488775740382835, −8.789999847702375993444720543543, −7.82417676035817458562471207559, −6.37206292281746949105673281199, −5.52964110576726671359675751837, −4.15465608214738534476864113060, −2.85191708908294916138026113073, −0.71581401436188315410771316539, 0.889149048579573550153199036303, 2.55086593283578715974973612680, 3.78714699434622072932626696166, 4.83011641578188884756396293243, 6.64827221822876579579642667310, 7.31845106563899283674616778496, 9.098197829815283171845775062726, 10.11062277623278274528032938725, 10.86900080513678351497317100220, 12.06332562697268935288209643589

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