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Properties

Label 2-126-63.16-c1-0-2
Degree $2$
Conductor $126$
Sign $0.823 - 0.566i$
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  + (0.5 + 0.866i)2-s + (−1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 3·5-s − 1.73i·6-s + (2.5 + 0.866i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (1.5 + 2.59i)10-s − 3·11-s + (1.49 − 0.866i)12-s + (0.5 + 0.866i)13-s + (0.500 + 2.59i)14-s + (−4.5 − 2.59i)15-s + (−0.5 − 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.34·5-s − 0.707i·6-s + (0.944 + 0.327i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (0.474 + 0.821i)10-s − 0.904·11-s + (0.433 − 0.250i)12-s + (0.138 + 0.240i)13-s + (0.133 + 0.694i)14-s + (−1.16 − 0.670i)15-s + (−0.125 − 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.823 - 0.566i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.823 - 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13563 + 0.352890i\)
\(L(\frac12)\) \(\approx\) \(1.13563 + 0.352890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 - 3T + 5T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 9T + 23T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (5.5 + 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.67041368915416936220025578945, −12.62836641332770141744593562549, −11.54718669175052799593195179595, −10.50659863752942918898004167627, −9.202746126331890823711891182369, −7.82423426144249283702094556522, −6.70163750038223861705109779247, −5.52734244289732304957434074852, −4.96588314051139918582978027048, −2.13636692247167841516222100035, 1.84956783432629104318704861358, 4.06702539239372604993027940264, 5.44079678921541417689612264012, 6.00345354899038989724305118790, 7.969708175379174987148766416288, 9.699408972551220269475491540593, 10.24193581406914219907586937559, 11.11503787621961102647179601502, 12.17371929257572711384110050821, 13.24681378013587923248407282181

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