Properties

Label 2-126-63.58-c1-0-3
Degree $2$
Conductor $126$
Sign $0.381 + 0.924i$
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  − 2-s + (−1.64 + 0.545i)3-s + 4-s + (−0.794 − 1.37i)5-s + (1.64 − 0.545i)6-s + (1.23 − 2.33i)7-s − 8-s + (2.40 − 1.79i)9-s + (0.794 + 1.37i)10-s + (0.794 − 1.37i)11-s + (−1.64 + 0.545i)12-s + (2.40 − 4.16i)13-s + (−1.23 + 2.33i)14-s + (2.05 + 1.82i)15-s + 16-s + (−2.69 − 4.67i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.949 + 0.314i)3-s + 0.5·4-s + (−0.355 − 0.615i)5-s + (0.671 − 0.222i)6-s + (0.468 − 0.883i)7-s − 0.353·8-s + (0.801 − 0.597i)9-s + (0.251 + 0.434i)10-s + (0.239 − 0.414i)11-s + (−0.474 + 0.157i)12-s + (0.667 − 1.15i)13-s + (−0.331 + 0.624i)14-s + (0.530 + 0.472i)15-s + 0.250·16-s + (−0.654 − 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $0.381 + 0.924i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1/2),\ 0.381 + 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.459259 - 0.307363i\)
\(L(\frac12)\) \(\approx\) \(0.459259 - 0.307363i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.64 - 0.545i)T \)
7 \( 1 + (-1.23 + 2.33i)T \)
good5 \( 1 + (0.794 + 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.794 + 1.37i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.40 + 4.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.69 + 4.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.54 - 6.14i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.150 + 0.260i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.13 - 7.16i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.71T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.66T + 47T^{2} \)
53 \( 1 + (-2.44 - 4.23i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.47T + 59T^{2} \)
61 \( 1 + 4.47T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 12.7T + 71T^{2} \)
73 \( 1 + (-8.02 - 13.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 8.38T + 79T^{2} \)
83 \( 1 + (-1.18 - 2.04i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.712 - 1.23i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.87484094954525194593663051091, −11.97082810644464250127551134373, −10.84732959791900234598302120257, −10.39670005005889916484766415281, −8.938586019222316464060645460913, −7.894191538808791936014529306411, −6.64343629647277429914394841098, −5.29011304663618827340426332134, −3.92107866396784158616243738204, −0.873986433272503384728038232416, 2.03750631602677229208003221670, 4.46608122365669305639871803248, 6.18356623052483934234293234208, 6.88601815630164515905870459924, 8.244284997232272599270867137285, 9.343063590666177901406344973970, 10.83955254478111696525142261941, 11.29599807704208171125261324161, 12.18388711779839018787106036372, 13.36279010228431881942096846163

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