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Properties

Label 2-126-63.4-c3-0-6
Degree $2$
Conductor $126$
Sign $0.880 - 0.473i$
Analytic cond. $7.43424$
Root an. cond. $2.72658$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)2-s + (−5.19 + 0.108i)3-s + (−1.99 − 3.46i)4-s − 14.9·5-s + (−5.00 + 9.10i)6-s + (14.0 + 12.0i)7-s − 7.99·8-s + (26.9 − 1.12i)9-s + (−14.9 + 25.8i)10-s + 58.7·11-s + (10.7 + 17.7i)12-s + (−21.3 + 36.9i)13-s + (34.9 − 12.3i)14-s + (77.5 − 1.61i)15-s + (−8 + 13.8i)16-s + (6.88 − 11.9i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.999 + 0.0208i)3-s + (−0.249 − 0.433i)4-s − 1.33·5-s + (−0.340 + 0.619i)6-s + (0.760 + 0.649i)7-s − 0.353·8-s + (0.999 − 0.0416i)9-s + (−0.471 + 0.817i)10-s + 1.61·11-s + (0.258 + 0.427i)12-s + (−0.455 + 0.788i)13-s + (0.666 − 0.235i)14-s + (1.33 − 0.0278i)15-s + (−0.125 + 0.216i)16-s + (0.0981 − 0.170i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.880 - 0.473i$
Analytic conductor: \(7.43424\)
Root analytic conductor: \(2.72658\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :3/2),\ 0.880 - 0.473i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.00974 + 0.254143i\)
\(L(\frac12)\) \(\approx\) \(1.00974 + 0.254143i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + 1.73i)T \)
3 \( 1 + (5.19 - 0.108i)T \)
7 \( 1 + (-14.0 - 12.0i)T \)
good5 \( 1 + 14.9T + 125T^{2} \)
11 \( 1 - 58.7T + 1.33e3T^{2} \)
13 \( 1 + (21.3 - 36.9i)T + (-1.09e3 - 1.90e3i)T^{2} \)
17 \( 1 + (-6.88 + 11.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-55.0 - 95.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + 147.T + 1.21e4T^{2} \)
29 \( 1 + (-105. - 182. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-33.4 - 57.9i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-196. - 340. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-188. + 325. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (103. + 180. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (49.8 - 86.3i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-55.4 + 96.0i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-115. - 199. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (417. - 722. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-48.4 - 83.8i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 637.T + 3.57e5T^{2} \)
73 \( 1 + (-227. + 394. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-135. + 234. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (351. + 608. i)T + (-2.85e5 + 4.95e5i)T^{2} \)
89 \( 1 + (150. + 260. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-361. - 626. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.15064057218167672635717805517, −12.00213482667469196335294421622, −11.45827464502538695064063305360, −10.15422521576991110257023105946, −8.845115115736526527023641108279, −7.45749821636586729420408834528, −6.13000996538759402682780126777, −4.70858848002967155378579915604, −3.84214335743756591967442720213, −1.40646778932963417753632069191, 0.63625072681856205500166236946, 3.95314115583400933300028584588, 4.65455390145363017251650438679, 6.18821824305640646445333647701, 7.34555840168909260013688272451, 8.034795436532125021079754904235, 9.724031350592561826283937462521, 11.26779800217379866853039231786, 11.67457278498863805572275575636, 12.62547949268565428066472002182

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