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Properties

Label 2-126-63.59-c1-0-2
Degree $2$
Conductor $126$
Sign $0.733 - 0.679i$
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 + (−1.73 − 0.0288i)3-s + (0.499 + 0.866i)4-s + 2.28·5-s + (−1.48 − 0.890i)6-s + (1.21 + 2.34i)7-s + 0.999i·8-s + (2.99 + 0.100i)9-s + (1.97 + 1.14i)10-s + 1.09i·11-s + (−0.840 − 1.51i)12-s + (−5.91 − 3.41i)13-s + (−0.117 + 2.64i)14-s + (−3.95 − 0.0659i)15-s + (−0.5 + 0.866i)16-s + (3.35 − 5.81i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.999 − 0.0166i)3-s + (0.249 + 0.433i)4-s + 1.02·5-s + (−0.606 − 0.363i)6-s + (0.461 + 0.887i)7-s + 0.353i·8-s + (0.999 + 0.0333i)9-s + (0.624 + 0.360i)10-s + 0.329i·11-s + (−0.242 − 0.437i)12-s + (−1.64 − 0.947i)13-s + (−0.0314 + 0.706i)14-s + (−1.02 − 0.0170i)15-s + (−0.125 + 0.216i)16-s + (0.814 − 1.41i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.733 - 0.679i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.733 - 0.679i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18770 + 0.465516i\)
\(L(\frac12)\) \(\approx\) \(1.18770 + 0.465516i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 + 0.0288i)T \)
7 \( 1 + (-1.21 - 2.34i)T \)
good5 \( 1 - 2.28T + 5T^{2} \)
11 \( 1 - 1.09iT - 11T^{2} \)
13 \( 1 + (5.91 + 3.41i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.35 + 5.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.47 - 1.43i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 3.90iT - 23T^{2} \)
29 \( 1 + (-1.59 + 0.923i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.75 - 1.01i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.57 - 6.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.45 - 4.25i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.74 + 6.48i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.40 + 5.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.222 + 0.128i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.971 - 1.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.15 - 0.665i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.54 + 4.41i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 0.233iT - 71T^{2} \)
73 \( 1 + (-5.89 - 3.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.63 + 6.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.91 - 5.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.99 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.13 + 2.38i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.40550747433682864455880089483, −12.32005230620211138630750244814, −11.90265883663398575525168530482, −10.38516849722562862816623890315, −9.563948286368833592772546452590, −7.83113989879515204254290566498, −6.60922575258977110561880133165, −5.41875133781415360320275906563, −4.94280085334466735864866620617, −2.42769359931582346975380916083, 1.77592278759005883717732465332, 4.16276542170888396471089587100, 5.28422348012485399375956561786, 6.33279081531095227354843243952, 7.48242326572795634279396358603, 9.579773819757478878163644157016, 10.33303397689669097096167040057, 11.22093827957626383300274516487, 12.28060526165706229227140015962, 13.14682453111436212130775551009

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