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Properties

Label 2-126-9.5-c2-0-11
Degree $2$
Conductor $126$
Sign $-0.807 + 0.589i$
Analytic cond. $3.43325$
Root an. cond. $1.85290$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (−1.41 − 2.64i)3-s + (0.999 − 1.73i)4-s + (−3.60 − 2.08i)5-s + (−3.60 − 2.24i)6-s + (−1.32 − 2.29i)7-s − 2.82i·8-s + (−5.01 + 7.47i)9-s − 5.89·10-s + (−5.60 + 3.23i)11-s + (−5.99 − 0.202i)12-s + (7.36 − 12.7i)13-s + (−3.24 − 1.87i)14-s + (−0.421 + 12.4i)15-s + (−2.00 − 3.46i)16-s − 18.6i·17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.470 − 0.882i)3-s + (0.249 − 0.433i)4-s + (−0.721 − 0.416i)5-s + (−0.600 − 0.373i)6-s + (−0.188 − 0.327i)7-s − 0.353i·8-s + (−0.557 + 0.830i)9-s − 0.589·10-s + (−0.509 + 0.294i)11-s + (−0.499 − 0.0168i)12-s + (0.566 − 0.980i)13-s + (−0.231 − 0.133i)14-s + (−0.0280 + 0.832i)15-s + (−0.125 − 0.216i)16-s − 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.807 + 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.807 + 0.589i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.807 + 0.589i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.397126 - 1.21732i\)
\(L(\frac12)\) \(\approx\) \(0.397126 - 1.21732i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 + (1.41 + 2.64i)T \)
7 \( 1 + (1.32 + 2.29i)T \)
good5 \( 1 + (3.60 + 2.08i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (5.60 - 3.23i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-7.36 + 12.7i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 18.6iT - 289T^{2} \)
19 \( 1 - 15.6T + 361T^{2} \)
23 \( 1 + (-17.1 - 9.88i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (9.30 - 5.37i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-11.1 + 19.3i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 70.8T + 1.36e3T^{2} \)
41 \( 1 + (7.29 + 4.20i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-24.9 - 43.2i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (62.5 - 36.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 80.3iT - 2.80e3T^{2} \)
59 \( 1 + (22.8 + 13.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-32.7 - 56.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (11.9 - 20.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 111. iT - 5.04e3T^{2} \)
73 \( 1 - 13.3T + 5.32e3T^{2} \)
79 \( 1 + (59.8 + 103. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-10.0 + 5.81i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 83.5iT - 7.92e3T^{2} \)
97 \( 1 + (84.0 + 145. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−12.94701801589616468194172926463, −11.71205452507747099936699963460, −11.16668095433490756959845958694, −9.798821965366039500306424899589, −8.074937509060295419378345808291, −7.25828346066407614303404245684, −5.83723324497955421949129968317, −4.71044314024530183786658951680, −2.96379943975587903198619440617, −0.802139226034390117433585145818, 3.24015144742142599720982437356, 4.31135607493739069099575986148, 5.64484898974510347317097489776, 6.70642862159318185631976295317, 8.150898128110779513095102228408, 9.352553846344196089783856497576, 10.78017780692860239356337527992, 11.43530398174233843685927475334, 12.40772203054104744626346257815, 13.66321058561861023979764346301

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