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Properties

Label 2-126-21.5-c1-0-3
Degree $2$
Conductor $126$
Sign $0.985 - 0.168i$
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 + (0.499 + 0.866i)4-s + (2.09 − 3.62i)5-s + (−1.62 + 2.09i)7-s + 0.999i·8-s + (3.62 − 2.09i)10-s + (−2.59 + 1.5i)11-s + 2.44i·13-s + (−2.44 + 0.999i)14-s + (−0.5 + 0.866i)16-s + (−0.507 − 0.878i)17-s + (−0.878 − 0.507i)19-s + 4.18·20-s − 3·22-s + (−3.67 − 2.12i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.935 − 1.61i)5-s + (−0.612 + 0.790i)7-s + 0.353i·8-s + (1.14 − 0.661i)10-s + (−0.783 + 0.452i)11-s + 0.679i·13-s + (−0.654 + 0.267i)14-s + (−0.125 + 0.216i)16-s + (−0.123 − 0.213i)17-s + (−0.201 − 0.116i)19-s + 0.935·20-s − 0.639·22-s + (−0.766 − 0.442i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.985 - 0.168i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.985 - 0.168i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49602 + 0.126689i\)
\(L(\frac12)\) \(\approx\) \(1.49602 + 0.126689i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (1.62 - 2.09i)T \)
good5 \( 1 + (-2.09 + 3.62i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (0.507 + 0.878i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.878 + 0.507i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.67 + 2.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 1.24iT - 29T^{2} \)
31 \( 1 + (-4.86 + 2.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.12 - 7.13i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.02T + 41T^{2} \)
43 \( 1 - 8.24T + 43T^{2} \)
47 \( 1 + (0.507 - 0.878i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.07 + 0.621i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.76 + 9.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.12 - 2.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + (-7.24 + 4.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.62 + 9.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.16T + 83T^{2} \)
89 \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.76iT - 97T^{2} \)
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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

−13.25292213632619626975872722678, −12.63617714891846620471241645384, −11.86885112365780663082114627898, −10.02760840387882736476277039588, −9.123919097793103516996065733477, −8.187042968845198272548302864760, −6.44279159329338704180310005479, −5.45091027992580206218923318806, −4.50966455895037023794726880306, −2.28204504192451370250960903204, 2.53910575083662099604526767520, 3.63075997645600839621479756175, 5.65278745410887970585731553620, 6.52254269220323586442557272963, 7.63394240889213259180219884345, 9.676187988778515871465835758364, 10.52820495678704976121793752481, 10.92055016454385686452644191908, 12.53573651810426380156954457905, 13.65014528747896657549649463703

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